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<p><strong>Description:</strong> Differentiates between independent and dependent events as it pertains to probability, covering applications like coin flips, the distribution of birthdays, hashing, and cryptography.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="390">The</span> <span m="510">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1660">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2430">Creative</span> <span m="2830">Commons</span> <span m="3230">license.</span> <span m="4340">Your</span> <span m="4530">support</span> <span m="5030">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high</span> <span m="7930">quality</span> <span m="8450">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation</span> <span m="11990">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13130">materials</span> <span m="13660">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14780">courses,</span> <span m="15900">visit</span> <span m="16110">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="23224">TOM LEIGHTON:</span> <span m="23700">Today</span> <span m="24220">we're</span> <span m="24360">going</span> <span m="24470">to</span> <span m="24560">talk</span> <span m="25030">about</span> <span m="25710">the</span> <span m="25810">concept</span> <span m="27080">of</span> <span m="27170">independence.</span> <span m="29560">In</span> <span m="29860">probability,</span> <span m="30940">we</span> <span m="31100">say</span> <span m="31560">that</span> <span m="36660">an</span> <span m="36830">event</span> <span m="37490">A</span> <span m="41760">is</span> <span m="42060">independent</span> <span m="50510">of</span> <span m="50800">an event</span> <span m="51120">B</span> <span m="57710">if</span> <span m="57930">one</span> <span m="58170">of</span> <span m="58260">two</span> <span m="58460">conditions</span> <span m="59060">hold.</span> <span m="60210">First,</span> <span m="62210">if</span> <span m="62430">the</span> <span m="62530">probability</span> <span m="63100">of</span> <span m="63180">A</span> <span m="63430">given</span> <span m="63760">B</span> <span m="64910">is</span> <span m="65129">just</span> <span m="65650">the</span> <span m="65730">same</span> <span m="66160">as</span> <span m="66270">the</span> <span m="66360">probability</span> <span m="66930">of</span> <span m="67020">A</span> <span m="68910">or</span> <span m="70320">if</span> <span m="71650">B</span> <span m="71930">can't</span> <span m="72270">happen,</span> <span m="72940">namely</span> <span m="73250">the</span> <span m="73350">probability</span> <span m="73930">of</span> <span m="74000">B</span> <span m="74770">is</span> <span m="74970">0.</span> </p>
<p><span m="76470">In</span> <span m="76600">other</span> <span m="76790">words,</span> <span m="77700">A</span> <span m="77940">is</span> <span m="78080">independent</span> <span m="78600">of</span> <span m="78730">B</span> <span m="79710">if</span> <span m="79950">knowing</span> <span m="80500">that</span> <span m="80670">B</span> <span m="80870">happened</span> <span m="82450">doesn't</span> <span m="82870">change</span> <span m="84060">the</span> <span m="84170">probability</span> <span m="84820">that</span> <span m="85010">A</span> <span m="85160">is</span> <span m="85310">going</span> <span m="85430">to</span> <span m="85490">happen.</span> <span m="86060">So</span> <span m="86950">knowing</span> <span m="88830">that</span> <span m="88900">this</span> <span m="89070">event</span> <span m="89300">occurs</span> <span m="89720">doesn't</span> <span m="90060">influence</span> <span m="90510">the</span> <span m="90580">probability</span> <span m="91070">that</span> <span m="91200">A</span> <span m="91370">occurs.</span> <span m="92460">And</span> <span m="92680">there's</span> <span m="92810">a</span> <span m="92890">special</span> <span m="93320">case</span> <span m="93780">where</span> <span m="93880">they're</span> <span m="94030">independent</span> <span m="94640">because</span> <span m="95400">you</span> <span m="95700">know</span> <span m="96015">that</span> <span m="96330">B</span> <span m="96550">can't</span> <span m="96820">happen.</span> <span m="97820">If</span> <span m="97930">the</span> <span m="98030">probability</span> <span m="98590">of</span> <span m="98660">B</span> <span m="99130">happening</span> <span m="99550">is</span> <span m="99710">0,</span> <span m="101020">then</span> <span m="101260">everything</span> <span m="101630">is</span> <span m="101730">independent</span> <span m="102370">of</span> <span m="102520">B.</span> </p>
<p><span m="104271">Now,</span> <span m="104770">the</span> <span m="104920">typical</span> <span m="105350">example</span> <span m="106410">that</span> <span m="106550">gets</span> <span m="106810">used</span> <span m="107260">is</span> <span m="107380">when</span> <span m="107530">you</span> <span m="107630">flip</span> <span m="107990">two</span> <span m="108190">coins.</span> <span m="113970">So</span> <span m="114160">say</span> <span m="114410">we</span> <span m="114570">flip</span> <span m="115230">two</span> <span m="115750">fair,</span> <span m="118590">independent</span> <span m="119230">coins.</span> <span m="128020">And</span> <span m="128320">let's</span> <span m="128590">let</span> <span m="129419">B</span> <span m="130639">be</span> <span m="130789">the</span> <span m="130949">event</span> <span m="133180">that</span> <span m="133310">the</span> <span m="133420">first</span> <span m="133930">coin</span> <span m="138740">is</span> <span m="139620">heads</span> <span m="141090">and</span> <span m="141650">that</span> <span m="142040">means</span> <span m="142530">that</span> <span m="142930">the</span> <span m="143020">probability</span> <span m="144400">of</span> <span m="144570">B</span> <span m="144850">happening</span> <span m="145930">is</span> <span m="146110">1/2,</span> <span m="147320">because</span> <span m="147680">we've</span> <span m="147860">assumed</span> <span m="148170">it's</span> <span m="148310">a</span> <span m="148360">fair</span> <span m="148690">coin,</span> <span m="150040">and</span> <span m="150260">we'll</span> <span m="150360">let</span> <span m="150610">A</span> <span m="151150">be</span> <span m="151340">the</span> <span m="151520">event</span> <span m="152130">that</span> <span m="152270">the</span> <span m="152370">second</span> <span m="152860">coin</span> <span m="155580">comes</span> <span m="155880">out</span> <span m="156110">heads.</span> <span m="158120">So</span> <span m="158280">we</span> <span m="158390">know</span> <span m="158610">the</span> <span m="158710">probability</span> <span m="159240">of</span> <span m="159340">A</span> <span m="159670">is</span> <span m="159840">1/2</span> <span m="160490">because</span> <span m="160690">it's</span> <span m="160880">fair.</span> </p>
<p><span m="165180">And</span> <span m="165620">because</span> <span m="166810">they're</span> <span m="167020">independent,</span> <span m="170140">we</span> <span m="170300">can</span> <span m="170440">conclude</span> <span m="171060">that</span> <span m="171170">the</span> <span m="171260">probability</span> <span m="172980">of</span> <span m="173210">A</span> <span m="173560">given</span> <span m="173950">B</span> <span m="175460">is</span> <span m="175710">1/2,</span> <span m="176900">which</span> <span m="177140">is</span> <span m="177260">the</span> <span m="177350">probability</span> <span m="177940">of</span> <span m="178050">A.</span> <span m="179920">In</span> <span m="180050">other</span> <span m="180250">words,</span> <span m="181780">seeing</span> <span m="182250">the</span> <span m="182330">result</span> <span m="182830">of</span> <span m="182890">the</span> <span m="182960">second</span> <span m="183390">coin</span> <span m="183830">doesn't</span> <span m="184210">tell</span> <span m="184510">you</span> <span m="184720">anything</span> <span m="185060">about</span> <span m="185990">the</span> <span m="186100">result</span> <span m="186640">of</span> <span m="186760">the</span> <span m="186840">first</span> <span m="187150">coin.</span> </p>
<p><span m="189190">Now</span> <span m="189910">actually,</span> <span m="191540">when</span> <span m="191660">you</span> <span m="191760">flip</span> <span m="192050">two</span> <span m="192250">coins,</span> <span m="193300">it's</span> <span m="193490">not</span> <span m="193740">just</span> <span m="194250">always</span> <span m="194800">the</span> <span m="194920">case</span> <span m="195330">if</span> <span m="195420">they're</span> <span m="195600">independent.</span> <span m="197520">Can</span> <span m="197660">anybody</span> <span m="198080">think</span> <span m="198360">of</span> <span m="198440">an</span> <span m="198500">example</span> <span m="199060">where</span> <span m="199240">you</span> <span m="199360">can</span> <span m="199510">flip</span> <span m="199920">a</span> <span m="200000">pair</span> <span m="200280">of</span> <span m="200360">coins</span> <span m="201730">and</span> <span m="201910">they</span> <span m="202040">are</span> <span m="202230">dependent</span> <span m="203020">somehow,</span> <span m="203550">they're</span> <span m="203660">not</span> <span m="203960">independent?</span> <span m="205820">Yeah.</span> </p>
<p><span m="206270">AUDIENCE: Well,</span> <span m="206751">if</span> <span m="206911">you</span> <span m="206964">have</span> <span m="207017">to</span> <span m="207071">get</span> <span m="207232">two</span> <span m="207713">heads</span> <span m="208194">and</span> <span m="208675">two</span> <span m="209156">tails?</span> </p>
<p><span m="209640">TOM LEIGHTON: If</span> <span m="209770">you</span> <span m="209900">have</span> <span m="210400">to</span> <span m="210500">get</span> <span m="210840">two</span> <span m="211020">heads</span> <span m="211320">or</span> <span m="211430">two</span> <span m="211540">tails.</span> <span m="213500">Well,</span> <span m="213690">how</span> <span m="213890">would</span> <span m="214080">you</span> <span m="214230">have</span> <span m="214410">to</span> <span m="214910">get?</span> </p>
<p><span m="215200">AUDIENCE: The</span> <span m="215490">probability</span> <span m="215982">of</span> <span m="216474">getting</span> <span m="216966">two</span> <span m="217458">heads</span> <span m="219426">should</span> <span m="219918">be</span> <span m="220410">1/4</span> <span m="220902">[INAUDIBLE].</span> </p>
<p><span m="222378">TOM LEIGHTON:</span> <span m="222870">Well,</span> <span m="223270">then</span> <span m="223440">they</span> <span m="223495">would</span> <span m="223550">be</span> <span m="223700">independent</span> <span m="224250">in</span> <span m="224320">that</span> <span m="224550">case.</span> <span m="225220">Yeah.</span> </p>
<p><span m="225515">AUDIENCE:</span> <span m="225810">If</span> <span m="225930">you</span> <span m="226080">glue</span> <span m="226330">the</span> <span m="226490">coins</span> <span m="226650">together.</span> </p>
<p><span m="227100">TOM LEIGHTON:</span> <span m="227360">Yeah.</span> <span m="228150">I</span> <span m="228250">mean,</span> <span m="228370">this</span> <span m="228510">is</span> <span m="228610">a</span> <span m="228700">silly</span> <span m="229060">example,</span> <span m="229550">but</span> <span m="229680">I</span> <span m="229740">got</span> <span m="230290">two</span> <span m="230630">fair</span> <span m="230940">coins</span> <span m="231370">here.</span> <span m="233160">I</span> <span m="233300">could</span> <span m="234200">clip</span> <span m="234540">them</span> <span m="234720">together</span> <span m="236220">and</span> <span m="236370">now</span> <span m="236510">I</span> <span m="236640">flip</span> <span m="236950">them</span> <span m="238080">and</span> <span m="238250">odds</span> <span m="238460">are</span> <span m="238520">pretty</span> <span m="238750">good</span> <span m="239030">they're</span> <span m="239150">both</span> <span m="239620">going</span> <span m="239720">to</span> <span m="239760">be</span> <span m="239890">heads</span> <span m="240120">or</span> <span m="240220">both</span> <span m="240490">be</span> <span m="240630">tails.</span> <span m="241990">If</span> <span m="242140">you</span> <span m="242270">know</span> <span m="242610">what</span> <span m="242760">happened</span> <span m="243130">to</span> <span m="243200">the</span> <span m="243300">right</span> <span m="243570">coin,</span> <span m="244230">it</span> <span m="244680">will</span> <span m="244920">tell</span> <span m="245100">you</span> <span m="245230">what</span> <span m="245330">happened</span> <span m="245630">to</span> <span m="245670">the</span> <span m="245750">left</span> <span m="246030">coin.</span> </p>
<p><span m="247040">Now,</span> <span m="247160">that's</span> <span m="247370">a</span> <span m="247460">pretty</span> <span m="248080">contrived</span> <span m="248810">example,</span> <span m="250600">but</span> <span m="250870">it</span> <span m="250950">is</span> <span m="251200">illustrative</span> <span m="251950">of</span> <span m="252080">what</span> <span m="252250">happens</span> <span m="252740">in</span> <span m="252860">practice.</span> <span m="254000">In</span> <span m="254140">practice,</span> <span m="254750">we</span> <span m="255000">assume</span> <span m="255850">independence</span> <span m="257610">even</span> <span m="257990">though</span> <span m="258200">there</span> <span m="258370">can</span> <span m="258760">be</span> <span m="258910">subtle</span> <span m="259339">dependencies</span> <span m="260550">and</span> <span m="260680">this</span> <span m="260839">could</span> <span m="260950">lead</span> <span m="261149">to</span> <span m="261220">trouble.</span> <span m="261620">In</span> <span m="261690">fact,</span> <span m="261899">we're</span> <span m="261970">going</span> <span m="262050">to</span> <span m="262089">give</span> <span m="262200">a</span> <span m="262270">lot</span> <span m="262530">of</span> <span m="262610">examples</span> <span m="263650">where</span> <span m="263910">it</span> <span m="263960">leads</span> <span m="264190">to</span> <span m="264270">trouble</span> <span m="264590">today</span> <span m="265340">and</span> <span m="265570">also</span> <span m="266090">for</span> <span m="266220">the</span> <span m="266320">rest</span> <span m="266570">of</span> <span m="266610">the</span> <span m="266700">course.</span> <span m="267560">Because</span> <span m="267730">we're</span> <span m="267910">always</span> <span m="268320">going</span> <span m="268440">to</span> <span m="268510">want</span> <span m="268730">to</span> <span m="268820">assume</span> <span m="269200">independence</span> <span m="270390">and</span> <span m="270600">when</span> <span m="270770">we</span> <span m="270890">do,</span> <span m="271150">we're</span> <span m="271250">going</span> <span m="271330">to</span> <span m="271370">get</span> <span m="271560">very</span> <span m="271830">nice</span> <span m="272090">results,</span> <span m="273400">but</span> <span m="273560">things</span> <span m="273940">aren't</span> <span m="274290">always</span> <span m="274730">independent</span> <span m="275020">in</span> <span m="275310">practice</span> <span m="275820">and</span> <span m="275900">establishing</span> <span m="276530">independence</span> <span m="277240">is</span> <span m="277370">a</span> <span m="277420">hard</span> <span m="277690">thing</span> <span m="277860">to</span> <span m="277970">do.</span> </p>
<p><span m="279860">For</span> <span m="280030">that</span> <span m="280250">matter,</span> <span m="280510">while</span> <span m="280720">we're</span> <span m="280850">on</span> <span m="280940">the</span> <span m="281030">subject,</span> <span m="281930">we</span> <span m="282050">always</span> <span m="282250">talk</span> <span m="282430">about</span> <span m="282660">fair</span> <span m="282930">coins.</span> <span m="283450">You</span> <span m="283540">flip</span> <span m="283750">a</span> <span m="283820">coin</span> <span m="284220">and</span> <span m="284295">it's</span> <span m="284370">fair.</span> <span m="285650">You</span> <span m="285760">know,</span> <span m="286340">that's</span> <span m="286600">not</span> <span m="286760">always</span> <span m="287000">to</span> <span m="287200">either.</span> <span m="287450">There's</span> <span m="288660">actually</span> <span m="288920">a</span> <span m="288970">famous</span> <span m="289390">mathematician</span> <span m="290130">named</span> <span m="290740">Persi</span> <span m="291080">Diaconis</span> <span m="292710">who</span> <span m="292830">used</span> <span m="293100">to</span> <span m="293310">down</span> <span m="293900">the</span> <span m="294000">street</span> <span m="294210">at</span> <span m="294280">Harvard</span> <span m="295100">and</span> <span m="295760">he</span> <span m="295910">came</span> <span m="296230">and</span> <span m="296320">gave</span> <span m="296480">a</span> <span m="296540">talk</span> <span m="296920">one</span> <span m="297160">day</span> <span m="298070">at</span> <span m="298260">MIT</span> <span m="298720">in</span> <span m="298790">the</span> <span m="298870">math</span> <span m="299120">department</span> <span m="299610">and</span> <span m="299690">he's</span> <span m="299880">a</span> <span m="299960">probabalist.</span> <span m="300540">He</span> <span m="300760">does</span> <span m="300950">probability</span> <span m="301530">theory</span> <span m="302060">and</span> <span m="302380">is</span> <span m="302550">a</span> <span m="302730">very</span> <span m="302920">cool</span> <span m="303110">guy.</span> </p>
<p><span m="303830">And</span> <span m="304170">so</span> <span m="304820">he</span> <span m="305770">flipped</span> <span m="306220">a</span> <span m="306280">coin,</span> <span m="306930">got</span> <span m="307090">a</span> <span m="307130">quarter</span> <span m="307570">from</span> <span m="307700">somebody</span> <span m="307890">in</span> <span m="308000">the</span> <span m="308090">audience</span> <span m="308520">and</span> <span m="308680">flipped</span> <span m="309050">it</span> <span m="310180">and</span> <span m="310290">he</span> <span m="310400">flip</span> <span m="310680">that</span> <span m="310780">I</span> <span m="310870">think</span> <span m="311310">10</span> <span m="311760">or</span> <span m="311850">20</span> <span m="312200">straight</span> <span m="312580">times</span> <span m="313050">all</span> <span m="313360">the</span> <span m="313460">way</span> <span m="313700">to</span> <span m="313770">the</span> <span m="313900">roof,</span> <span m="315310">caught</span> <span m="315640">it,</span> <span m="315960">turned</span> <span m="316210">it</span> <span m="316280">over.</span> <span m="316900">Every</span> <span m="317200">time</span> <span m="317460">it</span> <span m="317530">was</span> <span m="317690">heads.</span> <span m="318812">And</span> <span m="319160">he</span> <span m="319225">goes,</span> <span m="319290">now</span> <span m="319420">what's</span> <span m="319660">the</span> <span m="319790">probability</span> <span m="320510">of</span> <span m="320570">that</span> <span m="320780">happening?</span> <span m="322190">Well,</span> <span m="322350">you</span> <span m="322430">know,</span> <span m="322620">it's</span> <span m="322960">1/2</span> <span m="323510">to</span> <span m="323620">the</span> <span m="323700">20th</span> <span m="324170">or</span> <span m="324220">whatever,</span> <span m="324800">not</span> <span m="325120">very</span> <span m="325370">likely.</span> <span m="326130">How</span> <span m="326540">could</span> <span m="326650">he</span> <span m="326760">always</span> <span m="327110">make</span> <span m="327250">it</span> <span m="327320">come</span> <span m="327465">out</span> <span m="327610">heads?</span> </p>
<p><span m="328730">Well,</span> <span m="329130">Persi</span> <span m="329490">was</span> <span m="329640">an</span> <span m="329700">unusual</span> <span m="330210">guy</span> <span m="330950">and</span> <span m="331140">in</span> <span m="331230">fact,</span> <span m="331620">he'd</span> <span m="331780">spent</span> <span m="332120">months</span> <span m="332820">in</span> <span m="332920">the</span> <span m="333010">strobe</span> <span m="333500">lab</span> <span m="335130">over</span> <span m="335350">at</span> <span m="335400">Harvard</span> <span m="336300">practicing</span> <span m="337480">to</span> <span m="337630">make</span> <span m="337810">it</span> <span m="337980">always</span> <span m="338480">rotate</span> <span m="339040">seven</span> <span m="339400">times,</span> <span m="341250">three</span> <span m="341590">of</span> <span m="341730">them</span> <span m="341780">on</span> <span m="341820">the</span> <span m="341860">way</span> <span m="342000">up,</span> <span m="342240">one</span> <span m="342390">at</span> <span m="342430">the</span> <span m="342530">top,</span> <span m="342810">and</span> <span m="342900">then</span> <span m="343020">three</span> <span m="343230">down.</span> <span m="343460">And</span> <span m="343590">he</span> <span m="343845">could</span> <span m="344100">actually</span> <span m="344180">see</span> <span m="345390">how</span> <span m="345540">many</span> <span m="345740">rotations</span> <span m="346010">it</span> <span m="346280">had</span> <span m="346410">done</span> <span m="346580">to</span> <span m="346630">make</span> <span m="346800">sure</span> <span m="347070">it</span> <span m="347130">was</span> <span m="347260">seven,</span> <span m="348420">so</span> <span m="348590">it</span> <span m="348670">always</span> <span m="349210">came</span> <span m="349450">out</span> <span m="349620">heads.</span> </p>
<p><span m="350950">Now,</span> <span m="351350">he</span> <span m="351580">is</span> <span m="351690">an</span> <span m="351750">unusual</span> <span m="352170">fellow.</span> <span m="352560">He</span> <span m="352950">was</span> <span m="354020">1</span> <span m="354500">of</span> <span m="354640">10</span> <span m="354980">people</span> <span m="355340">in</span> <span m="355400">the</span> <span m="355500">world</span> <span m="356250">that</span> <span m="356360">could</span> <span m="356520">do</span> <span m="356660">a</span> <span m="356740">perfect</span> <span m="357310">shuffle</span> <span m="357840">reliably</span> <span m="358600">on</span> <span m="358700">a</span> <span m="358760">deck</span> <span m="359000">of</span> <span m="359090">cards</span> <span m="360660">and</span> <span m="360750">that's</span> <span m="360920">a</span> <span m="360960">very</span> <span m="361380">hard</span> <span m="361590">thing</span> <span m="361730">to</span> <span m="361800">do.</span> <span m="361940">He</span> <span m="362060">said</span> <span m="362330">he</span> <span m="362430">had</span> <span m="362470">to</span> <span m="362510">practice</span> <span m="362970">8</span> <span m="363170">hours</span> <span m="363480">a</span> <span m="363570">day</span> <span m="364270">for</span> <span m="364400">over</span> <span m="364580">six</span> <span m="364860">months</span> <span m="366330">to</span> <span m="366410">be</span> <span m="366620">able</span> <span m="366780">to</span> <span m="366870">do</span> <span m="367080">it</span> <span m="367190">every</span> <span m="367440">time.</span> <span m="367800">In</span> <span m="368080">fact,</span> <span m="368160">he</span> <span m="368270">gave</span> <span m="368380">another</span> <span m="368630">talk</span> <span m="368900">at</span> <span m="368950">MIT</span> <span m="369390">where</span> <span m="369465">he</span> <span m="369540">came</span> <span m="369830">in</span> <span m="369930">and</span> <span m="370060">he</span> <span m="370610">made</span> <span m="370860">magic</span> <span m="371180">tricks,</span> <span m="371870">actually</span> <span m="372210">based</span> <span m="372460">on</span> <span m="372510">mathematics.</span> <span m="373520">And</span> <span m="374100">you</span> <span m="374260">would</span> <span m="374450">cut</span> <span m="374680">a</span> <span m="374720">deck,</span> <span m="375650">he</span> <span m="375830">would</span> <span m="375990">feel</span> <span m="376220">it</span> <span m="376450">like</span> <span m="376660">this</span> <span m="377440">and</span> <span m="377590">tell</span> <span m="377720">you</span> <span m="378090">where</span> <span m="378340">you</span> <span m="378460">cut,</span> <span m="378710">how</span> <span m="378870">many</span> <span m="379040">cards</span> <span m="379500">were</span> <span m="379570">in</span> <span m="379640">the</span> <span m="379770">part</span> <span m="380000">you</span> <span m="380080">picked</span> <span m="380350">up</span> <span m="381620">and</span> <span m="381770">then</span> <span m="381930">do</span> <span m="382090">his</span> <span m="382240">eight</span> <span m="382390">perfect</span> <span m="382800">shuffles,</span> <span m="383220">which</span> <span m="383390">is</span> <span m="383490">enough</span> <span m="383710">to</span> <span m="383960">return</span> <span m="384215">a</span> <span m="384470">normal</span> <span m="384770">52-card</span> <span m="385480">deck</span> <span m="385980">back</span> <span m="386390">to</span> <span m="386520">its</span> <span m="386660">original</span> <span m="387300">order.</span> </p>
<p><span m="388720">And</span> <span m="388820">then</span> <span m="388870">using</span> <span m="389360">this,</span> <span m="390440">he</span> <span m="390580">could</span> <span m="390680">play</span> <span m="390810">the</span> <span m="390920">game</span> <span m="391170">where</span> <span m="391320">pick</span> <span m="391540">any</span> <span m="391690">card,</span> <span m="392960">you</span> <span m="393090">stick</span> <span m="393370">it</span> <span m="393460">in,</span> <span m="394360">he</span> <span m="394520">feels</span> <span m="395290">where</span> <span m="395480">the</span> <span m="395610">card</span> <span m="395940">went,</span> <span m="396650">and</span> <span m="396800">then</span> <span m="396920">using</span> <span m="397160">mathematics,</span> <span m="397800">he</span> <span m="397900">could</span> <span m="398030">shuffle</span> <span m="398480">the</span> <span m="398600">deck</span> <span m="398780">eight</span> <span m="398960">times</span> <span m="399260">and</span> <span m="399360">make</span> <span m="399500">the</span> <span m="399570">card</span> <span m="399830">come</span> <span m="399990">out</span> <span m="400180">anywhere</span> <span m="400500">he</span> <span m="400620">wanted</span> <span m="401450">in</span> <span m="401580">the</span> <span m="401670">deck.</span> <span m="402670">So</span> <span m="403050">he</span> <span m="403110">had</span> <span m="403210">a</span> <span m="403260">lot</span> <span m="403480">going</span> <span m="403710">on</span> <span m="403850">upstairs</span> <span m="404310">too.</span> </p>
<p><span m="406000">He</span> <span m="406210">had</span> <span m="406340">an</span> <span m="406400">interesting</span> <span m="406720">life</span> <span m="406960">history.</span> <span m="407910">He</span> <span m="408080">ran</span> <span m="408420">away</span> <span m="408710">from</span> <span m="408910">home</span> <span m="409890">as</span> <span m="410130">a</span> <span m="410180">young</span> <span m="410430">child</span> <span m="410880">and</span> <span m="411165">joined</span> <span m="411450">the</span> <span m="411560">traveling</span> <span m="412100">circus.</span> <span m="414150">And</span> <span m="414300">then</span> <span m="414420">somehow</span> <span m="414980">from</span> <span m="415190">there,</span> <span m="416050">he</span> <span m="416170">joined</span> <span m="416500">the</span> <span m="416570">faculty</span> <span m="417030">at</span> <span m="417100">Harvard.</span> <span m="417580">You</span> <span m="417870">know,</span> <span m="418060">there's</span> <span m="418220">an</span> <span m="418380">amazing</span> <span m="419000">story.</span> </p>
<p><span m="421190">And</span> <span m="421400">actually</span> <span m="421710">your</span> <span m="421970">story</span> <span m="422180">about</span> <span m="422430">Persi</span> <span m="422897">is</span> <span m="424300">he</span> <span m="424490">was</span> <span m="424650">the</span> <span m="424760">first</span> <span m="425230">guy</span> <span m="425510">to</span> <span m="425590">get</span> <span m="425800">kicked</span> <span m="426050">out</span> <span m="426190">of</span> <span m="426380">casinos</span> <span m="427020">for</span> <span m="427160">card</span> <span m="427480">counting.</span> <span m="428060">He</span> <span m="428440">figured</span> <span m="428730">that</span> <span m="428900">out</span> <span m="429070">way</span> <span m="429260">before</span> <span m="429980">the</span> <span m="430090">MIT</span> <span m="430590">team</span> <span m="430900">and</span> <span m="430960">the</span> <span m="431040">movie</span> <span m="431450"><i>21.</i></span> <span m="432970">Down</span> <span m="433130">in</span> <span m="433290">Puerto</span> <span m="433670">Rico,</span> <span m="434050">he</span> <span m="434190">used</span> <span m="434330">to</span> <span m="434400">play</span> <span m="434720">and</span> <span m="434960">then</span> <span m="435060">they</span> <span m="435170">finally</span> <span m="435550">figured</span> <span m="435860">him</span> <span m="436070">out</span> <span m="436250">and</span> <span m="436470">he</span> <span m="436610">got</span> <span m="436820">booted.</span> </p>
<p><span m="439470">So</span> <span m="439950">back</span> <span m="440190">to</span> <span m="440280">independence,</span> <span m="442600">let's</span> <span m="442840">do</span> <span m="443630">another</span> <span m="444230">picture</span> <span m="444550">example.</span> <span m="446230">Say</span> <span m="446490">that</span> <span m="446630">my</span> <span m="446770">sample</span> <span m="447230">space</span> <span m="447820">looks</span> <span m="448060">like</span> <span m="448290">this</span> <span m="449720">and</span> <span m="449900">I've</span> <span m="449990">got</span> <span m="450220">two</span> <span m="450410">events,</span> <span m="451560">A</span> <span m="452480">and</span> <span m="453400">B</span> <span m="453730">and</span> <span m="453810">they</span> <span m="453900">look</span> <span m="454170">like</span> <span m="454380">this,</span> <span m="454650">so</span> <span m="454740">they're</span> <span m="455110">dis-joined.</span> <span m="456456">Are</span> <span m="456840">A</span> <span m="457060">and</span> <span m="457430">B</span> <span m="457670">independent?</span> <span m="460340">No.</span> <span m="461410">In</span> <span m="461520">fact</span> <span m="461990">what</span> <span m="462270">is</span> <span m="462530">the</span> <span m="462640">probability</span> <span m="463310">of</span> <span m="463400">A</span> <span m="463570">given</span> <span m="463850">B</span> <span m="466590">as</span> <span m="466800">I've</span> <span m="466920">drawn</span> <span m="467390">it?</span> </p>
<p><span m="467780">AUDIENCE:</span> <span m="468170">0.</span> </p>
<p><span m="468560">TOM LEIGHTON:</span> <span m="468950">0.</span> <span m="469940">Because</span> <span m="470150">if</span> <span m="470550">B</span> <span m="470810">occurs,</span> <span m="472050">you're</span> <span m="472280">outside</span> <span m="472780">of</span> <span m="472880">A.</span> <span m="474371">And</span> <span m="474870">so</span> <span m="475120">this</span> <span m="475510">does</span> <span m="475690">not</span> <span m="475930">equal</span> <span m="476075">the</span> <span m="476220">probability</span> <span m="476800">of</span> <span m="476870">A</span> <span m="477175">as</span> <span m="477480">long</span> <span m="477700">as</span> <span m="477790">it's</span> <span m="477960">not</span> <span m="478410">0.</span> <span m="479860">So</span> <span m="480130">disjoint</span> <span m="480880">events</span> <span m="483200">don't</span> <span m="483700">imply</span> <span m="484010">that</span> <span m="484320">they're</span> <span m="484710">independent.</span> </p>
<p><span m="492730">Now,</span> <span m="493420">what's</span> <span m="493700">the</span> <span m="493810">picture</span> <span m="494150">look</span> <span m="494490">like</span> <span m="495510">for</span> <span m="495670">them</span> <span m="495840">to</span> <span m="495910">be</span> <span m="496050">independent?</span> <span m="496630">What</span> <span m="496735">is</span> <span m="496840">the</span> <span m="496960">right</span> <span m="497500">picture</span> <span m="497800">to</span> <span m="497900">draw</span> <span m="498230">here?</span> <span m="499926">So</span> <span m="500310">I</span> <span m="500410">got</span> <span m="500580">my</span> <span m="500700">sample</span> <span m="501150">space</span> <span m="502400">and</span> <span m="502570">say</span> <span m="502860">I</span> <span m="502880">make</span> <span m="503300">this</span> <span m="503680">half</span> <span m="504080">the</span> <span m="504170">sample</span> <span m="504580">space</span> <span m="504930">be</span> <span m="505210">A.</span> <span m="506930">Well,</span> <span m="507230">then</span> <span m="507560">B</span> <span m="508070">to</span> <span m="508140">be</span> <span m="508270">independent,</span> <span m="509880">would</span> <span m="510450">look</span> <span m="510650">something--</span> <span m="511110">I</span> <span m="511295">didn't</span> <span m="511480">quite</span> <span m="511860">draw</span> <span m="512130">it.</span> <span m="512175">I</span> <span m="512220">actually</span> <span m="512580">have</span> <span m="512770">it</span> <span m="512850">be</span> <span m="512960">50-50.</span> </p>
<p><span m="515309">So</span> <span m="516370">if</span> <span m="517140">a</span> <span m="517429">is</span> <span m="517700">50%</span> <span m="518539">of</span> <span m="518669">S,</span> <span m="519270">like</span> <span m="519470">this</span> <span m="519710">half,</span> <span m="521100">then</span> <span m="521860">for</span> <span m="522010">A</span> <span m="522159">to</span> <span m="522240">be</span> <span m="522320">independent</span> <span m="522890">of</span> <span m="522980">B,</span> <span m="523382">A</span> <span m="523784">intersect</span> <span m="524186">B,</span> <span m="525790">this</span> <span m="525990">part,</span> <span m="527360">has</span> <span m="527660">to</span> <span m="527730">be</span> <span m="527860">50%</span> <span m="528490">of</span> <span m="528550">B.</span> <span m="529850">Because</span> <span m="530540">the</span> <span m="530640">probability</span> <span m="531860">of</span> <span m="532040">A</span> <span m="532390">given</span> <span m="532680">B</span> <span m="533920">must</span> <span m="534190">equal</span> <span m="534520">the</span> <span m="534610">probability</span> <span m="535190">of</span> <span m="535270">A</span> <span m="536500">to</span> <span m="536720">be</span> <span m="536820">independent.</span> <span m="538170">So</span> <span m="538210">this</span> <span m="538440">would</span> <span m="538540">be</span> <span m="538670">a</span> <span m="538750">picture</span> <span m="539170">where</span> <span m="539480">they</span> <span m="539660">are</span> <span m="539870">independent.</span> </p>
<p><span m="543560">Now,</span> <span m="543650">independent</span> <span m="544180">events</span> <span m="544570">are</span> <span m="544660">really</span> <span m="544950">nice</span> <span m="545210">to</span> <span m="545330">work</span> <span m="545620">with</span> <span m="545970">and</span> <span m="546605">in</span> <span m="546890">part</span> <span m="547200">because</span> <span m="547410">they</span> <span m="547490">have</span> <span m="547600">a</span> <span m="547640">very</span> <span m="548000">simple</span> <span m="548540">rule</span> <span m="549560">for</span> <span m="549720">computing</span> <span m="550170">the</span> <span m="550260">probability</span> <span m="551080">of</span> <span m="551220">an</span> <span m="551330">intersection</span> <span m="552060">of</span> <span m="552190">events</span> <span m="555070">and</span> <span m="555160">it's</span> <span m="555300">called</span> <span m="555640">the</span> <span m="555720">product</span> <span m="556210">rule</span> <span m="557860">for</span> <span m="558290">independent</span> <span m="558730">events.</span> <span m="574150">And</span> <span m="574670">that</span> <span m="574970">says</span> <span m="575470">that</span> <span m="575590">if</span> <span m="575830">A</span> <span m="577230">is</span> <span m="577700">independent</span> <span m="578290">of</span> <span m="578370">B,</span> <span m="584620">then</span> <span m="585110">the</span> <span m="585200">probability</span> <span m="585990">of</span> <span m="586215">A</span> <span m="586440">and</span> <span m="586950">B</span> <span m="587240">or</span> <span m="587410">A</span> <span m="587580">intersect</span> <span m="588110">B</span> <span m="591000">is</span> <span m="591280">just</span> <span m="591560">the</span> <span m="591640">product</span> <span m="593160">of</span> <span m="593630">their</span> <span m="593780">probabilities</span> <span m="594680">separately,</span> <span m="595880">the</span> <span m="596010">probability</span> <span m="596690">of</span> <span m="596770">A</span> <span m="596940">times</span> <span m="597280">the</span> <span m="597360">probability</span> <span m="597930">of</span> <span m="598040">B.</span> </p>
<p><span m="599290">So</span> <span m="599430">let's</span> <span m="599680">prove</span> <span m="600030">this.</span> <span m="605630">And</span> <span m="605810">there's</span> <span m="605990">two</span> <span m="606180">cases,</span> <span m="606850">depending</span> <span m="607230">on</span> <span m="607410">whether</span> <span m="607670">or</span> <span m="607710">not</span> <span m="608560">B</span> <span m="608770">can</span> <span m="608900">happen,</span> <span m="609390">if</span> <span m="609550">the</span> <span m="609640">probability</span> <span m="610150">of</span> <span m="610220">B</span> <span m="610470">is</span> <span m="610620">0</span> <span m="610870">or</span> <span m="610970">not.</span> <span m="612490">So</span> <span m="612830">case</span> <span m="613120">1</span> <span m="617120">is</span> <span m="617740">B</span> <span m="617930">can't</span> <span m="618210">happen.</span> <span m="618880">The</span> <span m="618950">probability</span> <span m="619390">of</span> <span m="619490">B</span> <span m="619690">is</span> <span m="619860">0.</span> <span m="622220">In</span> <span m="622350">this</span> <span m="622590">case,</span> <span m="625090">what's</span> <span m="625300">the</span> <span m="625390">probability</span> <span m="625930">of</span> <span m="625965">A</span> <span m="626000">and</span> <span m="626490">B?</span> <span m="630680">B</span> <span m="630840">can't</span> <span m="631110">happen.</span> <span m="632330">0.</span> </p>
<p><span m="633170">If</span> <span m="633390">B</span> <span m="633530">can't</span> <span m="633810">happen,</span> <span m="634190">then</span> <span m="634970">they</span> <span m="635060">both</span> <span m="635330">can't.</span> <span m="636620">You</span> <span m="636720">can't</span> <span m="636950">have</span> <span m="637150">both</span> <span m="637360">of</span> <span m="637440">them</span> <span m="637560">happening</span> <span m="638800">and</span> <span m="639020">that</span> <span m="639210">equals</span> <span m="641070">the</span> <span m="641140">probability</span> <span m="641630">of</span> <span m="641710">A</span> <span m="643700">times</span> <span m="644020">the</span> <span m="644100">probability</span> <span m="644525">of</span> <span m="644950">B</span> <span m="647080">because</span> <span m="647290">the</span> <span m="647360">probability</span> <span m="647780">of</span> <span m="647850">B</span> <span m="647990">is</span> <span m="648150">0.</span> <span m="648670">So</span> <span m="648840">that</span> <span m="648980">case</span> <span m="649230">works.</span> </p>
<p><span m="653170">Case</span> <span m="653460">2</span> <span m="655680">is</span> <span m="656040">the</span> <span m="656130">probability</span> <span m="656710">of</span> <span m="656790">B</span> <span m="657150">is</span> <span m="657290">bigger</span> <span m="657550">than</span> <span m="657710">0.</span> <span m="661720">In</span> <span m="661850">that</span> <span m="662060">case,</span> <span m="662500">we</span> <span m="662610">have</span> <span m="664730">the</span> <span m="664810">probability</span> <span m="665400">of</span> <span m="665445">A</span> <span m="665490">and</span> <span m="665860">B,</span> <span m="668020">A</span> <span m="668190">intersect</span> <span m="668660">B,</span> <span m="669766">well,</span> <span m="670140">from</span> <span m="670220">the</span> <span m="670300">definition,</span> <span m="670980">is</span> <span m="671080">the</span> <span m="671170">probability</span> <span m="671690">of</span> <span m="671770">B</span> <span m="673390">times</span> <span m="673850">the</span> <span m="673930">probability</span> <span m="674630">of</span> <span m="674820">A</span> <span m="675080">given</span> <span m="675390">B.</span> <span m="676900">We</span> <span m="677000">did</span> <span m="677150">that</span> <span m="677320">last</span> <span m="677620">time.</span> <span m="679270">And</span> <span m="679480">by</span> <span m="679640">independence,</span> <span m="681160">this</span> <span m="681390">is</span> <span m="681500">just</span> <span m="681810">the</span> <span m="681900">probability</span> <span m="682400">of</span> <span m="682480">A</span> <span m="684370">because</span> <span m="684630">A</span> <span m="684880">is</span> <span m="685155">independent</span> <span m="685292">of</span> <span m="685430">B,</span> <span m="686630">so</span> <span m="686860">we're</span> <span m="687020">done.</span> </p>
<p><span m="693280">In</span> <span m="693370">fact,</span> <span m="694360">many</span> <span m="694750">texts</span> <span m="695270">will</span> <span m="695470">define</span> <span m="697610">independence</span> <span m="698420">by</span> <span m="698640">this</span> <span m="698840">product</span> <span m="699230">rule.</span> <span m="700460">Many</span> <span m="700750">texts</span> <span m="701100">will</span> <span m="701250">say</span> <span m="701730">that</span> <span m="701990">A</span> <span m="702320">and</span> <span m="702690">B</span> <span m="702890">are</span> <span m="702990">independent</span> <span m="703650">if</span> <span m="703870">this</span> <span m="704110">is</span> <span m="704240">true.</span> <span m="706320">And</span> <span m="706520">it's</span> <span m="706650">equivalent,</span> <span m="707310">it</span> <span m="707420">turns</span> <span m="707700">out.</span> <span m="708070">We</span> <span m="708180">won't</span> <span m="708360">prove</span> <span m="708610">that</span> <span m="708780">here,</span> <span m="709100">but</span> <span m="709760">if</span> <span m="709880">you</span> <span m="709970">use</span> <span m="710240">this</span> <span m="710460">as</span> <span m="710540">the</span> <span m="710640">definition,</span> <span m="711810">then</span> <span m="711980">you</span> <span m="712090">can</span> <span m="712220">derive</span> <span m="712620">our</span> <span m="712830">definition</span> <span m="713460">as</span> <span m="713560">a</span> <span m="713630">result.</span> <span m="714170">So</span> <span m="714320">this</span> <span m="714550">is</span> <span m="714660">an</span> <span m="714750">equivalent</span> <span m="715300">definition</span> <span m="715820">of</span> <span m="715900">independence.</span> </p>
<p><span m="719960">Another</span> <span m="720240">nice</span> <span m="720580">fact</span> <span m="721730">about</span> <span m="722460">independent</span> <span m="723040">events</span> <span m="723570">is</span> <span m="723940">that</span> <span m="725210">it's</span> <span m="725420">a</span> <span m="725660">symmetric</span> <span m="726310">relationship.</span> <span m="732210">It's</span> <span m="732630">called</span> <span m="732870">the</span> <span m="732960">symmetry</span> <span m="733470">of</span> <span m="733550">independence.</span> <span m="746330">That</span> <span m="746560">says</span> <span m="746970">that</span> <span m="747360">if</span> <span m="747590">they</span> <span m="747895">A</span> <span m="748960">is</span> <span m="749560">independent</span> <span m="750150">of</span> <span m="750270">B,</span> <span m="754540">then</span> <span m="754730">the</span> <span m="754820">reverse</span> <span m="755230">is</span> <span m="755340">true.</span> <span m="756440">B</span> <span m="757460">is</span> <span m="758020">independent</span> <span m="758590">of</span> <span m="758690">A.</span> <span m="763048">Now,</span> <span m="763500">we</span> <span m="763610">won't</span> <span m="764020">prove</span> <span m="764490">that.</span> <span m="766240">It's</span> <span m="766410">actually</span> <span m="766780">easier</span> <span m="767100">to</span> <span m="767420">see</span> <span m="767540">that</span> <span m="767660">it's</span> <span m="767780">true</span> <span m="768590">if</span> <span m="768760">this</span> <span m="768970">were</span> <span m="769030">the</span> <span m="769140">definition</span> <span m="771550">of</span> <span m="771710">independence</span> <span m="772380">because</span> <span m="773510">A</span> <span m="773700">intersect</span> <span m="774200">B</span> <span m="774370">is</span> <span m="774490">the</span> <span m="774570">same</span> <span m="774810">as</span> <span m="774940">B</span> <span m="775130">intersect</span> <span m="775530">A</span> <span m="777006">and</span> <span m="777500">multiplication</span> <span m="778280">is</span> <span m="778540">commutative.</span> <span m="779750">So</span> <span m="779930">it's</span> <span m="780040">easier</span> <span m="780320">to</span> <span m="780410">see</span> <span m="780660">it</span> <span m="780740">if</span> <span m="780820">we</span> <span m="780920">had</span> <span m="780980">used</span> <span m="781230">that</span> <span m="781440">definition.</span> </p>
<p><span m="783980">So</span> <span m="785660">because</span> <span m="786080">of</span> <span m="786180">this</span> <span m="786360">we</span> <span m="786520">often</span> <span m="786750">just</span> <span m="786900">say</span> <span m="787150">A</span> <span m="787400">and</span> <span m="787700">B</span> <span m="787860">are</span> <span m="787950">independent</span> <span m="788770">because</span> <span m="788940">it</span> <span m="789010">doesn't</span> <span m="789260">matter</span> <span m="789540">which</span> <span m="789760">order</span> <span m="790360">you're</span> <span m="790490">taking</span> <span m="790790">them</span> <span m="790950">in.</span> <span m="793450">All</span> <span m="793530">right,</span> <span m="793640">any</span> <span m="793790">questions</span> <span m="795090">about</span> <span m="795260">the</span> <span m="795340">definition</span> <span m="797420">so</span> <span m="797550">far?</span> <span m="799171">All</span> <span m="799610">right.</span> <span m="799970">Let's</span> <span m="800420">do</span> <span m="800900">some</span> <span m="801100">examples.</span> </p>
<p><span m="814590">Let's</span> <span m="814680">say</span> <span m="814830">I</span> <span m="814920">have</span> <span m="815220">two</span> <span m="815780">independent</span> <span m="816680">fair</span> <span m="817050">coins.</span> <span m="827730">And</span> <span m="828010">I'm</span> <span m="828110">going</span> <span m="828230">to</span> <span m="828270">have</span> <span m="828820">the</span> <span m="828990">event</span> <span m="829510">A</span> <span m="831440">be</span> <span m="832970">the</span> <span m="833050">situation</span> <span m="833960">when</span> <span m="834170">the</span> <span m="834280">coins</span> <span m="835060">match,</span> <span m="836330">both</span> <span m="836590">heads,</span> <span m="836890">both</span> <span m="837100">tails.</span> <span m="841700">And</span> <span m="842010">B</span> <span m="842290">is</span> <span m="842430">going</span> <span m="842560">to</span> <span m="842620">be</span> <span m="842750">the</span> <span m="842900">event</span> <span m="844570">that</span> <span m="844740">the</span> <span m="844830">first</span> <span m="845220">coin</span> <span m="845600">is</span> <span m="845895">heads.</span> <span m="852420">And</span> <span m="852820">I</span> <span m="852870">want</span> <span m="853130">to</span> <span m="853180">know,</span> <span m="853500">are</span> <span m="853635">A</span> <span m="853770">and</span> <span m="854260">B</span> <span m="854940">independent?</span> <span m="858300">Are</span> <span m="858430">those</span> <span m="858810">independent</span> <span m="859530">events?</span> </p>
<p><span m="862995">Well,</span> <span m="864890">what's</span> <span m="865090">the</span> <span m="865200">first</span> <span m="866840">answer</span> <span m="867200">to</span> <span m="867260">this?</span> <span m="867650">I</span> <span m="867690">mean,</span> <span m="869040">A</span> <span m="869610">is</span> <span m="869960">event</span> <span m="870260">the</span> <span m="870350">coins</span> <span m="870720">match.</span> <span m="871290">B</span> <span m="871555">tells</span> <span m="871820">me</span> <span m="871890">what</span> <span m="872010">the</span> <span m="872090">first</span> <span m="872350">coin</span> <span m="872620">was.</span> <span m="873670">So</span> <span m="873790">the</span> <span m="873870">first</span> <span m="874160">inclination</span> <span m="874760">here</span> <span m="875100">is</span> <span m="875190">that</span> <span m="875440">these</span> <span m="875650">are</span> <span m="875710">dependent</span> <span m="876340">events</span> <span m="878370">because</span> <span m="879380">I</span> <span m="879500">know</span> <span m="879740">something</span> <span m="879960">about</span> <span m="880170">the</span> <span m="880250">first</span> <span m="880540">coin,</span> <span m="880785">so</span> <span m="881030">that</span> <span m="881200">might</span> <span m="881470">tell</span> <span m="881640">me</span> <span m="881760">something</span> <span m="882090">about</span> <span m="882380">the</span> <span m="882450">probability</span> <span m="883020">they</span> <span m="883170">match.</span> <span m="884510">There</span> <span m="884680">could</span> <span m="884880">be</span> <span m="884980">some</span> <span m="885190">dependence</span> <span m="885690">here.</span> </p>
<p><span m="887080">Now,</span> <span m="887250">in</span> <span m="887340">fact,</span> <span m="887750">because</span> <span m="887850">it's</span> <span m="887960">set</span> <span m="888190">up,</span> <span m="889010">they're</span> <span m="889110">independent</span> <span m="890110">and</span> <span m="890240">we</span> <span m="890360">can</span> <span m="890510">check</span> <span m="890780">that</span> <span m="890980">by</span> <span m="891090">just</span> <span m="892330">doing</span> <span m="892560">the</span> <span m="892650">calculation,</span> <span m="893390">computing</span> <span m="893880">the</span> <span m="893960">probability</span> <span m="894500">of</span> <span m="894580">A</span> <span m="894720">given</span> <span m="894970">B.</span> <span m="898440">Maybe</span> <span m="898640">I</span> <span m="898690">can</span> <span m="898820">do</span> <span m="898920">that.</span> <span m="899220">I'll</span> <span m="899336">do</span> <span m="899453">that</span> <span m="899570">here.</span> <span m="900910">The</span> <span m="901570">probability</span> <span m="902170">of</span> <span m="902340">A</span> <span m="903460">given</span> <span m="903780">B</span> <span m="906360">is,</span> <span m="907410">well,</span> <span m="909320">the</span> <span m="909430">condition</span> <span m="910890">that</span> <span m="911040">they're</span> <span m="911200">going</span> <span m="911310">to</span> <span m="911370">match</span> <span m="911970">given</span> <span m="912190">that</span> <span m="912310">the</span> <span m="912400">first</span> <span m="912690">point</span> <span m="912870">is</span> <span m="913000">heads</span> <span m="913440">means</span> <span m="914370">it's</span> <span m="914550">the</span> <span m="914630">same</span> <span m="914960">as</span> <span m="915110">the</span> <span m="915200">second</span> <span m="915570">coin</span> <span m="915820">being</span> <span m="916040">heads.</span> <span m="918280">This</span> <span m="918450">is</span> <span m="918510">the</span> <span m="918600">probability</span> <span m="919280">the</span> <span m="919340">second</span> <span m="919780">coin</span> <span m="922168">is</span> <span m="922640">heads</span> <span m="925260">and</span> <span m="925390">that's</span> <span m="925630">just</span> <span m="925890">1/2</span> <span m="927380">because</span> <span m="927530">it's</span> <span m="927680">a</span> <span m="927730">fair</span> <span m="927990">coin</span> <span m="930020">and</span> <span m="930240">independent</span> <span m="930740">of</span> <span m="930800">the</span> <span m="930880">first</span> <span m="931230">one.</span> </p>
<p><span m="931440">Now,</span> <span m="931940">the</span> <span m="932030">probability</span> <span m="934170">of</span> <span m="935350">A,</span> <span m="936900">by</span> <span m="937100">itself,</span> <span m="937520">the</span> <span m="937620">events</span> <span m="937920">the</span> <span m="938020">coins</span> <span m="938370">match,</span> <span m="938820">what's</span> <span m="939090">that?</span> <span m="939860">How</span> <span m="940000">much</span> <span m="940190">is</span> <span m="940280">that?</span> <span m="948340">What's</span> <span m="948540">the</span> <span m="948630">probability</span> <span m="949120">the</span> <span m="949220">coins</span> <span m="949580">match?</span> </p>
<p><span m="952640">AUDIENCE:</span> <span m="952950">[INAUDIBLE].</span> </p>
<p><span m="953945">TOM LEIGHTON:</span> <span m="954440">1/4</span> <span m="954890">plus</span> <span m="955210">1/4.</span> <span m="955680">I've</span> <span m="955900">got</span> <span m="956030">1/4</span> <span m="956160">chance</span> <span m="956480">of</span> <span m="956560">heads,</span> <span m="956820">heads</span> <span m="958020">1/4</span> <span m="958450">chance</span> <span m="958740">of</span> <span m="958830">tails,</span> <span m="959200">tails,</span> <span m="959560">so</span> <span m="959730">it's</span> <span m="959870">1/2,</span> <span m="962100">so</span> <span m="962930">it</span> <span m="963000">works</span> <span m="963280">out.</span> <span m="963670">The</span> <span m="964920">probability</span> <span m="965560">of</span> <span m="965660">A</span> <span m="965800">given</span> <span m="966050">B</span> <span m="966430">equals</span> <span m="967570">the</span> <span m="967670">probability</span> <span m="968100">of</span> <span m="968220">A.</span> <span m="968340">They're</span> <span m="968440">both</span> <span m="968770">1/2.</span> <span m="970200">So</span> <span m="971640">A</span> <span m="972020">and</span> <span m="972330">B</span> <span m="972540">are</span> <span m="972640">independent</span> <span m="973270">events</span> <span m="974770">because</span> <span m="974920">that's</span> <span m="975160">just</span> <span m="975340">the</span> <span m="975430">definition</span> <span m="976400">even</span> <span m="976640">though</span> <span m="976725">it</span> <span m="976810">looked</span> <span m="977180">like</span> <span m="977370">there</span> <span m="977460">might</span> <span m="977670">have</span> <span m="977750">been</span> <span m="977830">some</span> <span m="977970">dependence</span> <span m="978500">lurking</span> <span m="978850">around</span> <span m="979190">here.</span> </p>
<p><span m="981170">Now,</span> <span m="981760">this</span> <span m="981950">example</span> <span m="982410">that</span> <span m="982500">I</span> <span m="982570">just</span> <span m="982780">did</span> <span m="982910">is</span> <span m="982990">a</span> <span m="983070">little</span> <span m="983350">misleading.</span> <span m="984630">The</span> <span m="985130">intuition</span> <span m="986120">they</span> <span m="986290">probably</span> <span m="986920">are</span> <span m="986980">dependent</span> <span m="987950">actually</span> <span m="988360">is</span> <span m="988490">good</span> <span m="988660">intuition</span> <span m="989140">in</span> <span m="989220">this</span> <span m="989380">case</span> <span m="990970">because</span> <span m="991330">if</span> <span m="991490">I</span> <span m="991590">don't</span> <span m="992060">have</span> <span m="992370">fair</span> <span m="992680">coins,</span> <span m="994970">they</span> <span m="995130">are</span> <span m="995320">dependent.</span> <span m="997520">All</span> <span m="997650">right.</span> <span m="997850">So</span> <span m="998020">in</span> <span m="998090">particular,</span> <span m="999230">let's</span> <span m="999340">look</span> <span m="999490">at</span> <span m="999560">what</span> <span m="999700">happens</span> <span m="1000220">if</span> <span m="1001150">the</span> <span m="1001640">probability</span> <span m="1002220">of</span> <span m="1002295">a</span> <span m="1002370">heads</span> <span m="1003680">is</span> <span m="1003870">p</span> <span m="1005220">and</span> <span m="1005400">the</span> <span m="1005470">probability</span> <span m="1006030">of</span> <span m="1006130">tails</span> <span m="1007840">is</span> <span m="1007990">1</span> <span m="1008240">minus</span> <span m="1008610">p</span> <span m="1008900">for</span> <span m="1008990">both</span> <span m="1009280">coins.</span> </p>
<p><span m="1012070">So</span> <span m="1012220">let's</span> <span m="1012420">compute</span> <span m="1014200">the</span> <span m="1014600">probability</span> <span m="1015220">of</span> <span m="1015290">A</span> <span m="1015460">given</span> <span m="1015730">B.</span> <span m="1018260">What</span> <span m="1018430">is</span> <span m="1018610">it</span> <span m="1019510">in</span> <span m="1019630">this</span> <span m="1019790">case?</span> <span m="1022600">Well,</span> <span m="1022900">it's</span> <span m="1023165">the</span> <span m="1023253">probability</span> <span m="1023341">the</span> <span m="1023430">second</span> <span m="1023720">coin</span> <span m="1023950">is</span> <span m="1024050">heads.</span> <span m="1024380">What's</span> <span m="1024579">that?</span> <span m="1028319">p</span> <span m="1030450">because</span> <span m="1030560">both</span> <span m="1030750">of</span> <span m="1030940">them</span> <span m="1031109">are</span> <span m="1031190">heads</span> <span m="1031440">with</span> <span m="1031540">probability,</span> <span m="1032060">p.</span> <span m="1032700">They're</span> <span m="1032859">independent</span> <span m="1033390">still.</span> <span m="1034109">The</span> <span m="1034230">two</span> <span m="1034380">coins</span> <span m="1034690">are</span> <span m="1034750">independent.</span> </p>
<p><span m="1036130">And</span> <span m="1036250">now</span> <span m="1036390">let's</span> <span m="1036609">look</span> <span m="1036760">at</span> <span m="1036819">the</span> <span m="1036900">probability</span> <span m="1038369">that</span> <span m="1038440">the</span> <span m="1038530">coins</span> <span m="1038920">match.</span> <span m="1040859">Well,</span> <span m="1041150">it's</span> <span m="1041270">a</span> <span m="1041329">probability</span> <span m="1041869">of</span> <span m="1041940">heads,</span> <span m="1042210">heads</span> <span m="1042530">and</span> <span m="1042569">the</span> <span m="1042609">probability</span> <span m="1042970">of</span> <span m="1043040">tails,</span> <span m="1043400">tails.</span> <span m="1044520">Heads,</span> <span m="1044849">heads</span> <span m="1045960">is</span> <span m="1046150">p</span> <span m="1046339">times</span> <span m="1046740">p.</span> <span m="1048180">Tails,</span> <span m="1048680">tails</span> <span m="1049200">is</span> <span m="1049310">1</span> <span m="1049570">minus</span> <span m="1049970">p</span> <span m="1050290">squared.</span> <span m="1054050">So</span> <span m="1054300">to</span> <span m="1054570">independent,</span> <span m="1055450">I</span> <span m="1055540">need</span> <span m="1055950">this</span> <span m="1056250">to</span> <span m="1056320">equal</span> <span m="1056710">that</span> <span m="1057840">or</span> <span m="1058110">to</span> <span m="1058170">have</span> <span m="1059516">the</span> <span m="1059890">probability</span> <span m="1060370">of</span> <span m="1060540">B</span> <span m="1060710">be</span> <span m="1060880">0.</span> </p>
<p><span m="1062330">So</span> <span m="1063300">A</span> <span m="1063580">and</span> <span m="1063970">B</span> <span m="1066670">are</span> <span m="1066870">independent</span> <span m="1070000">if</span> <span m="1070240">and</span> <span m="1070340">only</span> <span m="1070730">if--</span> <span m="1072380">the</span> <span m="1072520">first</span> <span m="1072850">case</span> <span m="1073220">is</span> <span m="1074220">probability</span> <span m="1074880">B</span> <span m="1076206">is</span> <span m="1076570">0,</span> <span m="1077330">which</span> <span m="1077750">means</span> <span m="1078110">that</span> <span m="1078770">p</span> <span m="1079040">equals</span> <span m="1079310">0,</span> <span m="1082970">or</span> <span m="1083840">that</span> <span m="1084390">has</span> <span m="1084660">to</span> <span m="1084720">equal</span> <span m="1085040">this.</span> <span m="1088470">So</span> <span m="1088780">p</span> <span m="1089080">would</span> <span m="1089190">have</span> <span m="1089370">to</span> <span m="1089450">equal</span> <span m="1089810">1</span> <span m="1090110">minus</span> <span m="1092090">2p</span> <span m="1093690">plus</span> <span m="1093980">2p</span> <span m="1094390">squared,</span> <span m="1095770">just</span> <span m="1096850">square</span> <span m="1097130">that</span> <span m="1097380">out</span> <span m="1097590">there.</span> <span m="1099420">So</span> <span m="1099750">let's</span> <span m="1100010">solve</span> <span m="1100400">this.</span> <span m="1100750">That</span> <span m="1100970">happens</span> <span m="1101460">if</span> <span m="1101610">and</span> <span m="1101700">only</span> <span m="1102050">if</span> <span m="1103550">0</span> <span m="1104140">equals</span> <span m="1104580">1</span> <span m="1104910">minus</span> <span m="1105390">3p</span> <span m="1106590">plus</span> <span m="1107000">2p</span> <span m="1107420">squared.</span> <span m="1109630">That's</span> <span m="1110060">true</span> <span m="1110380">if</span> <span m="1110540">and</span> <span m="1110630">only</span> <span m="1110970">if</span> <span m="1112270">0</span> <span m="1112740">equals--</span> <span m="1113160">I</span> <span m="1113270">factor</span> <span m="1113760">this--</span> <span m="1114090">it's</span> <span m="1114340">1</span> <span m="1114650">minus</span> <span m="1114990">2p</span> <span m="1116460">times</span> <span m="1116680">1</span> <span m="1116870">minus</span> <span m="1117200">p</span> <span m="1118946">and</span> <span m="1119360">that's</span> <span m="1119600">if</span> <span m="1119750">and</span> <span m="1119840">only</span> <span m="1120170">if</span> <span m="1121280">p</span> <span m="1121500">is</span> <span m="1121630">1/2</span> <span m="1123460">or</span> <span m="1123710">p</span> <span m="1124060">is</span> <span m="1124490">1,</span> <span m="1126300">two</span> <span m="1126530">roots.</span> </p>
<p><span m="1128510">So</span> <span m="1128740">if</span> <span m="1128810">the</span> <span m="1128910">coins</span> <span m="1129330">are</span> <span m="1129440">always</span> <span m="1129790">heads,</span> <span m="1130690">they're</span> <span m="1130840">independent.</span> <span m="1131160">If</span> <span m="1131480">they're</span> <span m="1131630">always</span> <span m="1131980">tails,</span> <span m="1132690">the</span> <span m="1133090">events</span> <span m="1133460">are</span> <span m="1133530">independent</span> <span m="1134040">or</span> <span m="1134280">if</span> <span m="1134350">they're</span> <span m="1134480">fair</span> <span m="1134830">coins,</span> <span m="1136840">these</span> <span m="1137110">two</span> <span m="1137270">events</span> <span m="1137610">are</span> <span m="1137680">independent.</span> <span m="1138300">But</span> <span m="1138460">anything</span> <span m="1138920">else,</span> <span m="1140580">they're</span> <span m="1140740">not</span> <span m="1141250">independent</span> <span m="1141750">anymore.</span> <span m="1144280">Any</span> <span m="1144340">questions?</span> <span m="1146320">And</span> <span m="1146470">now</span> <span m="1146510">you</span> <span m="1146550">can</span> <span m="1146606">sort</span> <span m="1146663">of</span> <span m="1146720">see</span> <span m="1147140">if</span> <span m="1149380">the</span> <span m="1149690">coins</span> <span m="1149920">are</span> <span m="1149970">likely</span> <span m="1150310">to</span> <span m="1150380">be</span> <span m="1151320">tails</span> <span m="1151800">and</span> <span m="1151850">the</span> <span m="1151920">first</span> <span m="1152210">one</span> <span m="1152290">comes</span> <span m="1152520">up</span> <span m="1152650">heads,</span> <span m="1153360">that</span> <span m="1153580">should</span> <span m="1153830">influence</span> <span m="1154780">the</span> <span m="1154890">probability</span> <span m="1155350">the</span> <span m="1155450">coins</span> <span m="1155760">match.</span> <span m="1157361">It</span> <span m="1157740">should</span> <span m="1157990">change.</span> </p>
<p><span m="1161220">Questions?</span> <span m="1162530">All</span> <span m="1162620">right.</span> <span m="1162790">So</span> <span m="1162880">there's</span> <span m="1163020">a</span> <span m="1163070">nice</span> <span m="1163340">application</span> <span m="1163950">of</span> <span m="1164030">this</span> <span m="1164310">to</span> <span m="1164400">getting</span> <span m="1164730">an</span> <span m="1164810">edge</span> <span m="1165480">in</span> <span m="1165630">ultimate</span> <span m="1166050">Frisbee.</span> <span m="1167860">Now,</span> <span m="1169030">when</span> <span m="1169150">you're</span> <span m="1169240">playing</span> <span m="1169580">ultimate,</span> <span m="1170400">you've</span> <span m="1170520">got</span> <span m="1170650">to</span> <span m="1170710">decide</span> <span m="1171040">who</span> <span m="1171210">gets</span> <span m="1171680">the</span> <span m="1171750">Frisbee</span> <span m="1172090">first.</span> <span m="1173740">And</span> <span m="1174420">sometimes</span> <span m="1174850">you</span> <span m="1174920">don't</span> <span m="1175100">have</span> <span m="1175320">a</span> <span m="1175400">coin</span> <span m="1175960">to</span> <span m="1176050">flip,</span> <span m="1177140">call</span> <span m="1177370">heads</span> <span m="1177590">or</span> <span m="1177650">tails,</span> <span m="1178400">but</span> <span m="1178460">you</span> <span m="1178530">do</span> <span m="1178700">have</span> <span m="1178870">the</span> <span m="1178950">Frisbee.</span> </p>
<p><span m="1180860">Now,</span> <span m="1181030">you</span> <span m="1181160">could</span> <span m="1181290">flip</span> <span m="1181590">the</span> <span m="1181680">Frisbee</span> <span m="1182115">and</span> <span m="1182550">call</span> <span m="1183150">right</span> <span m="1183400">side</span> <span m="1183630">up</span> <span m="1183890">or</span> <span m="1184030">not,</span> <span m="1184950">but</span> <span m="1185080">the</span> <span m="1185170">problem</span> <span m="1185590">is</span> <span m="1185830">the</span> <span m="1185930">Frisbee</span> <span m="1186350">is</span> <span m="1186530">known</span> <span m="1187070">not</span> <span m="1187290">to</span> <span m="1187340">be</span> <span m="1187480">a</span> <span m="1187530">fair</span> <span m="1187790">coin.</span> <span m="1189020">When</span> <span m="1189090">you</span> <span m="1189170">toss</span> <span m="1189490">it</span> <span m="1189560">up</span> <span m="1189650">in</span> <span m="1189730">the</span> <span m="1189850">air,</span> <span m="1190050">it's</span> <span m="1190190">likely</span> <span m="1190720">to</span> <span m="1190800">wind</span> <span m="1191130">up</span> <span m="1191390">on,</span> <span m="1191700">I</span> <span m="1191750">guess,</span> <span m="1191940">the</span> <span m="1192350">curved</span> <span m="1192640">edge</span> <span m="1193170">down.</span> <span m="1193980">So</span> <span m="1194110">that</span> <span m="1194290">wouldn't</span> <span m="1194500">be</span> <span m="1194620">fair</span> <span m="1195160">to</span> <span m="1195250">call</span> <span m="1195480">heads</span> <span m="1195680">or</span> <span m="1195740">tails.</span> </p>
<p><span m="1197690">So</span> <span m="1197980">the</span> <span m="1198080">standard</span> <span m="1198440">solution</span> <span m="1198940">is</span> <span m="1199270">to</span> <span m="1199380">flip</span> <span m="1199680">the</span> <span m="1199940">two</span> <span m="1200240">Frisbees</span> <span m="1200730">at</span> <span m="1200800">the</span> <span m="1200880">same</span> <span m="1201170">time</span> <span m="1201620">or</span> <span m="1201750">one</span> <span m="1202080">Frisbee</span> <span m="1202450">twice</span> <span m="1203520">and</span> <span m="1203810">somebody</span> <span m="1204180">calls</span> <span m="1204620">same</span> <span m="1205190">or</span> <span m="1205360">different,</span> <span m="1207570">that</span> <span m="1208790">the</span> <span m="1208880">two</span> <span m="1209040">Frisbees</span> <span m="1209560">both</span> <span m="1209940">come</span> <span m="1210110">up</span> <span m="1210250">on</span> <span m="1210340">the</span> <span m="1210420">same</span> <span m="1210850">way</span> <span m="1211440">or</span> <span m="1211660">they</span> <span m="1211780">come</span> <span m="1211950">up</span> <span m="1212090">different</span> <span m="1212380">ways</span> <span m="1213750">and</span> <span m="1213900">then</span> <span m="1214040">if</span> <span m="1214110">you</span> <span m="1214220">called</span> <span m="1214510">it</span> <span m="1214640">right,</span> <span m="1215010">you</span> <span m="1215130">get</span> <span m="1215270">to</span> <span m="1215340">start</span> <span m="1215570">with</span> <span m="1215700">a</span> <span m="1215740">Frisbee.</span> <span m="1217000">And</span> <span m="1217140">the</span> <span m="1217280">idea</span> <span m="1217660">behind</span> <span m="1218080">this</span> <span m="1218460">is</span> <span m="1218700">that</span> <span m="1219145">that</span> <span m="1220650">simulates</span> <span m="1221080">a</span> <span m="1221130">fair</span> <span m="1221390">coin,</span> <span m="1222700">that</span> <span m="1223180">the</span> <span m="1223270">probability</span> <span m="1223770">that</span> <span m="1223920">they're</span> <span m="1224632">the</span> <span m="1224990">same</span> <span m="1225650">is</span> <span m="1225820">50-50.</span> </p>
<p><span m="1227970">What</span> <span m="1228090">do</span> <span m="1228130">you</span> <span m="1228200">think.</span> <span m="1228390">Is</span> <span m="1228470">that</span> <span m="1228610">a</span> <span m="1228660">fair</span> <span m="1228930">way</span> <span m="1229090">to</span> <span m="1229180">decide</span> <span m="1230290">who</span> <span m="1230410">starts</span> <span m="1230710">first?</span> <span m="1232300">Yeah.</span> </p>
<p><span m="1232560">AUDIENCE:</span> <span m="1233006">No.</span> </p>
<p><span m="1233452">TOM LEIGHTON:</span> <span m="1233900">No.</span> <span m="1234510">Yeah,</span> <span m="1235030">that's</span> <span m="1235240">right.</span> <span m="1235500">It's</span> <span m="1235680">not.</span> <span m="1237740">Now,</span> <span m="1238140">it</span> <span m="1238290">is</span> <span m="1239770">in</span> <span m="1239930">the</span> <span m="1240050">case</span> <span m="1240520">when</span> <span m="1242410">the</span> <span m="1242520">coin</span> <span m="1242800">was</span> <span m="1242950">fair,</span> <span m="1243590">but</span> <span m="1243710">we</span> <span m="1243800">know</span> <span m="1243940">the</span> <span m="1244050">Frisbee</span> <span m="1244330">is</span> <span m="1244400">not</span> <span m="1244610">fair.</span> <span m="1245610">And</span> <span m="1245720">in</span> <span m="1245830">fact,</span> <span m="1245875">you</span> <span m="1245920">can</span> <span m="1246220">see</span> <span m="1246620">this</span> <span m="1246900">from</span> <span m="1248470">this</span> <span m="1248720">probability.</span> <span m="1250040">This</span> <span m="1250250">is</span> <span m="1250340">the</span> <span m="1250440">probability</span> <span m="1251250">of</span> <span m="1251400">a</span> <span m="1251460">match,</span> <span m="1254680">which</span> <span m="1255590">is</span> <span m="1258360">fine</span> <span m="1258750">at</span> <span m="1258850">p</span> <span m="1259005">equal</span> <span m="1259160">1/2,</span> <span m="1260530">but</span> <span m="1260650">in</span> <span m="1260760">fact,</span> <span m="1261190">if</span> <span m="1261770">you</span> <span m="1261920">analyze</span> <span m="1262380">this</span> <span m="1262550">equation,</span> <span m="1263090">you</span> <span m="1263190">find</span> <span m="1263520">out</span> <span m="1263660">its</span> <span m="1263840">minimum</span> <span m="1264390">value</span> <span m="1266300">is</span> <span m="1266760">at</span> <span m="1266980">p</span> <span m="1267140">equals</span> <span m="1267410">1/2</span> <span m="1267950">and</span> <span m="1268250">as</span> <span m="1268380">p</span> <span m="1268770">starts</span> <span m="1269070">moving</span> <span m="1269320">away</span> <span m="1269660">from</span> <span m="1269830">1/2</span> <span m="1270200">towards</span> <span m="1270570">0</span> <span m="1270940">or</span> <span m="1271060">to</span> <span m="1271140">1,</span> <span m="1271860">it</span> <span m="1272010">gets</span> <span m="1272240">bigger.</span> <span m="1274520">And</span> <span m="1274700">we</span> <span m="1274820">know</span> <span m="1275240">that</span> <span m="1275450">for</span> <span m="1275560">Frisbees,</span> <span m="1275885">p</span> <span m="1276210">is</span> <span m="1276600">not</span> <span m="1276840">1/2.</span> <span m="1277610">This</span> <span m="1277870">means</span> <span m="1278510">that</span> <span m="1279190">the</span> <span m="1279290">probability</span> <span m="1279770">of</span> <span m="1279830">a</span> <span m="1279890">match</span> <span m="1280490">is</span> <span m="1280680">better</span> <span m="1281080">than</span> <span m="1281250">50%.</span> </p>
<p><span m="1283610">So</span> <span m="1283840">if</span> <span m="1283910">you're</span> <span m="1284020">ever</span> <span m="1284260">playing</span> <span m="1284540">ultimate,</span> <span m="1285020">always</span> <span m="1285370">call</span> <span m="1285600">same</span> <span m="1287040">because</span> <span m="1287270">you're</span> <span m="1287306">going</span> <span m="1287343">to</span> <span m="1287380">have</span> <span m="1287550">a</span> <span m="1287600">better</span> <span m="1287870">than</span> <span m="1287970">50-50</span> <span m="1288490">chance</span> <span m="1288860">of</span> <span m="1289120">getting</span> <span m="1289260">to</span> <span m="1289340">start</span> <span m="1289600">with</span> <span m="1289680">the</span> <span m="1289720">Frisbee.</span> <span m="1290720">It's</span> <span m="1290850">not</span> <span m="1291040">a</span> <span m="1291090">fair</span> <span m="1292120">example.</span> <span m="1295390">There</span> <span m="1295560">is</span> <span m="1295820">another</span> <span m="1296050">example</span> <span m="1296480">of</span> <span m="1296530">how</span> <span m="1296710">to</span> <span m="1296790">make</span> <span m="1297300">a</span> <span m="1297380">fair</span> <span m="1297700">coin</span> <span m="1298000">from</span> <span m="1299300">a</span> <span m="1299380">biased</span> <span m="1299800">coin</span> <span m="1300090">to</span> <span m="1300210">an</span> <span m="1300290">unbiased</span> <span m="1300790">coin</span> <span m="1300905">in</span> <span m="1301020">homework,</span> <span m="1301850">ways</span> <span m="1302230">of</span> <span m="1302370">doing</span> <span m="1302720">this</span> <span m="1303560">that</span> <span m="1303680">are</span> <span m="1303860">fair.</span> <span m="1304760">Because</span> <span m="1304960">often</span> <span m="1305230">you</span> <span m="1305310">have</span> <span m="1306480">biased</span> <span m="1306960">random</span> <span m="1307270">numbers</span> <span m="1307650">and</span> <span m="1307710">you</span> <span m="1307810">want</span> <span m="1307990">to</span> <span m="1308040">get</span> <span m="1308160">unbiased</span> <span m="1308760">or</span> <span m="1309440">maybe</span> <span m="1309640">you</span> <span m="1309710">got</span> <span m="1309830">a</span> <span m="1309870">fair</span> <span m="1310150">coin</span> <span m="1310920">and</span> <span m="1311040">you</span> <span m="1311120">want</span> <span m="1311310">to</span> <span m="1311350">make</span> <span m="1311580">something</span> <span m="1311830">that</span> <span m="1312300">comes</span> <span m="1312550">up</span> <span m="1312660">heads</span> <span m="1312890">with</span> <span m="1312960">probability</span> <span m="1313420">1/3.</span> <span m="1314570">How</span> <span m="1314830">do</span> <span m="1314890">you</span> <span m="1315020">actually</span> <span m="1315370">do</span> <span m="1315560">that</span> <span m="1315800">in</span> <span m="1315900">a</span> <span m="1315960">way</span> <span m="1316160">that</span> <span m="1316270">works?</span> <span m="1317720">Any</span> <span m="1317900">questions</span> <span m="1319550">on</span> <span m="1319660">that?</span> </p>
<p><span m="1324210">The</span> <span m="1324290">next</span> <span m="1324570">example</span> <span m="1325440">is</span> <span m="1325670">from</span> <span m="1326020">the</span> <span m="1326200">first</span> <span m="1327080">OJ</span> <span m="1327500">Simpson</span> <span m="1327980">trial.</span> <span m="1329190">How</span> <span m="1329370">many</span> <span m="1329550">people</span> <span m="1330050">here</span> <span m="1330230">know</span> <span m="1330410">who</span> <span m="1330620">OJ</span> <span m="1330870">Simpson</span> <span m="1331250">is?</span> <span m="1332930">OK,</span> <span m="1333120">so</span> <span m="1333240">he's</span> <span m="1333330">still</span> <span m="1333550">pretty</span> <span m="1333760">famous.</span> <span m="1335610">Now,</span> <span m="1336300">as</span> <span m="1336430">you</span> <span m="1336530">probably</span> <span m="1336890">know</span> <span m="1337080">then</span> <span m="1337240">he</span> <span m="1337340">was</span> <span m="1337510">a</span> <span m="1337570">famous</span> <span m="1338010">football</span> <span m="1338410">player.</span> <span m="1339040">Back</span> <span m="1339230">when</span> <span m="1339370">I</span> <span m="1339440">was</span> <span m="1339630">a</span> <span m="1339710">kid,</span> <span m="1340520">he</span> <span m="1340650">was</span> <span m="1342010">a</span> <span m="1342550">famous</span> <span m="1342810">college</span> <span m="1343210">player,</span> <span m="1343500">then</span> <span m="1343580">he</span> <span m="1343660">was</span> <span m="1343780">a</span> <span m="1343830">famous</span> <span m="1344220">pro</span> <span m="1344640">player</span> <span m="1346460">and</span> <span m="1346650">then</span> <span m="1346780">he</span> <span m="1346850">was</span> <span m="1346990">an</span> <span m="1347070">actor,</span> <span m="1347620">famous</span> <span m="1347970">actor.</span> </p>
<p><span m="1349100">And</span> <span m="1349180">then</span> <span m="1350840">he</span> <span m="1351320">was</span> <span m="1351490">accused</span> <span m="1351940">of</span> <span m="1352070">murdering</span> <span m="1352520">his</span> <span m="1352660">wife</span> <span m="1353630">in</span> <span m="1353750">a</span> <span m="1353820">gory</span> <span m="1354430">knifing</span> <span m="1355150">and</span> <span m="1355570">a</span> <span m="1355620">friend</span> <span m="1355980">of</span> <span m="1356050">his</span> <span m="1356180">wife's.</span> <span m="1357570">And</span> <span m="1358190">ultimately,</span> <span m="1358940">the</span> <span m="1359070">jury</span> <span m="1359480">found</span> <span m="1359695">him</span> <span m="1359910">not</span> <span m="1360220">guilty,</span> <span m="1361060">but</span> <span m="1361200">pretty</span> <span m="1361370">much</span> <span m="1361640">everybody</span> <span m="1362130">in</span> <span m="1362200">the</span> <span m="1362290">country</span> <span m="1362710">thought</span> <span m="1363020">he</span> <span m="1363150">did</span> <span m="1363330">it.</span> <span m="1363820">He</span> <span m="1363960">looked</span> <span m="1364530">really</span> <span m="1364750">guilty.</span> <span m="1366050">And</span> <span m="1366350">it</span> <span m="1366400">was</span> <span m="1366540">a</span> <span m="1366600">big</span> <span m="1367290">media</span> <span m="1367760">event,</span> <span m="1368400">one</span> <span m="1368560">of</span> <span m="1368720">the</span> <span m="1368890">first</span> <span m="1369210">big</span> <span m="1369410">trial</span> <span m="1369870">events</span> <span m="1370180">on</span> <span m="1370310">TV.</span> <span m="1371280">And</span> <span m="1371470">so</span> <span m="1371650">all</span> <span m="1371940">the</span> <span m="1372040">proceedings</span> <span m="1372520">were</span> <span m="1372635">on</span> <span m="1372750">TV</span> <span m="1373200">and</span> <span m="1373500">everybody</span> <span m="1374070">watched</span> <span m="1374390">them.</span> <span m="1374930">We'd</span> <span m="1375120">all</span> <span m="1375300">go</span> <span m="1375480">home</span> <span m="1375820">to</span> <span m="1375930">watch</span> <span m="1376230">the</span> <span m="1376360">OJ</span> <span m="1377240">hearing.</span> <span m="1377580">It</span> <span m="1377690">was</span> <span m="1377830">amazing.</span> </p>
<p><span m="1379930">Now,</span> <span m="1381620">during</span> <span m="1381960">the</span> <span m="1382100">indictment</span> <span m="1382580">proceedings,</span> <span m="1383270">there</span> <span m="1383430">was</span> <span m="1383570">a</span> <span m="1383620">huge</span> <span m="1384040">dispute</span> <span m="1384670">over</span> <span m="1384900">what</span> <span m="1385180">independence</span> <span m="1385920">was</span> <span m="1387140">and</span> <span m="1387300">does</span> <span m="1387480">it</span> <span m="1387590">matter.</span> <span m="1389650">The</span> <span m="1389810">issue</span> <span m="1390080">arose</span> <span m="1390560">when</span> <span m="1390630">the</span> <span m="1390720">prosecution</span> <span m="1392010">witness</span> <span m="1393240">claimed</span> <span m="1393810">that</span> <span m="1394140">only</span> <span m="1394410">1</span> <span m="1394750">in</span> <span m="1394920">200</span> <span m="1395490">Americans</span> <span m="1396740">had</span> <span m="1397020">a</span> <span m="1397080">certain</span> <span m="1397750">blood</span> <span m="1398050">type</span> <span m="1398730">that</span> <span m="1398900">matched</span> <span m="1399230">the</span> <span m="1399330">blood</span> <span m="1399620">type</span> <span m="1399900">found</span> <span m="1400420">at</span> <span m="1400580">the</span> <span m="1400670">scene</span> <span m="1400910">of</span> <span m="1400980">the</span> <span m="1401040">crime,</span> <span m="1401840">which</span> <span m="1402080">was</span> <span m="1402550">alleged</span> <span m="1402900">to</span> <span m="1402950">be</span> <span m="1403150">OJ's</span> <span m="1403540">blood.</span> <span m="1404450">And</span> <span m="1404710">this</span> <span m="1404860">was</span> <span m="1405110">during</span> <span m="1405330">the</span> <span m="1405440">indictment</span> <span m="1406030">and</span> <span m="1406150">back</span> <span m="1406380">then</span> <span m="1406610">DNA</span> <span m="1406980">tests</span> <span m="1407450">took</span> <span m="1407580">a</span> <span m="1407620">long</span> <span m="1407910">time</span> <span m="1408720">and</span> <span m="1408840">they</span> <span m="1408910">weren't</span> <span m="1409110">ready</span> <span m="1409350">yet.</span> <span m="1410590">And</span> <span m="1410930">the</span> <span m="1411020">witness</span> <span m="1411630">presented</span> <span m="1413080">the</span> <span m="1413200">following</span> <span m="1413710">facts</span> <span m="1413940">and</span> <span m="1414170">this</span> <span m="1414380">was</span> <span m="1414560">the</span> <span m="1415510">crime</span> <span m="1415890">lab</span> <span m="1416200">guy,</span> <span m="1417540">the</span> <span m="1417640">police</span> <span m="1418020">guy.</span> </p>
<p><span m="1432500">He</span> <span m="1432700">said</span> <span m="1433210">that</span> <span m="1433510">1</span> <span m="1433750">in</span> <span m="1433920">10</span> <span m="1434220">people,</span> <span m="1435830">roughly,</span> <span m="1437860">matched</span> <span m="1438310">type</span> <span m="1438740">O</span> <span m="1438960">blood.</span> <span m="1444000">And</span> <span m="1444410">that</span> <span m="1444560">1</span> <span m="1444830">in</span> <span m="1444940">5</span> <span m="1445360">people</span> <span m="1447440">matched</span> <span m="1448410">the</span> <span m="1448580">Rh</span> <span m="1449110">factor</span> <span m="1450000">positive.</span> <span m="1453910">And</span> <span m="1454310">that</span> <span m="1454480">1</span> <span m="1454710">in</span> <span m="1454810">4</span> <span m="1455130">people</span> <span m="1458040">match</span> <span m="1458620">a</span> <span m="1458710">certain</span> <span m="1459000">kind</span> <span m="1459180">of</span> <span m="1459310">marker,</span> <span m="1459820">which</span> <span m="1460610">I</span> <span m="1460720">don't</span> <span m="1460840">remember</span> <span m="1461060">what</span> <span m="1461230">it</span> <span m="1461330">was.</span> <span m="1461610">We'll</span> <span m="1461750">just</span> <span m="1461870">call</span> <span m="1461985">it</span> <span m="1462100">marker</span> <span m="1462550">XYZ,</span> <span m="1463950">some</span> <span m="1464150">other</span> <span m="1464840">factor</span> <span m="1465270">of</span> <span m="1465340">the</span> <span m="1465420">blood.</span> <span m="1466920">And</span> <span m="1467060">then</span> <span m="1467220">this</span> <span m="1467370">conclusion</span> <span m="1467980">was</span> <span m="1468520">that</span> <span m="1469800">this</span> <span m="1470010">means</span> <span m="1470330">that</span> <span m="1471000">1</span> <span m="1471290">in</span> <span m="1471430">200</span> <span m="1473370">match</span> <span m="1473740">all</span> <span m="1473930">three</span> <span m="1479350">factors.</span> </p>
<p><span m="1480720">And</span> <span m="1481660">this</span> <span m="1481870">seems</span> <span m="1482140">reasonable</span> <span m="1482580">because</span> <span m="1482835">there's</span> <span m="1483970">1/10</span> <span m="1484260">of</span> <span m="1484650">the</span> <span m="1484715">people</span> <span m="1484780">have</span> <span m="1485000">O,</span> <span m="1485880">if</span> <span m="1486200">15</span> <span m="1486320">of</span> <span m="1486673">them</span> <span m="1487380">have</span> <span m="1488200">positive</span> <span m="1488470">Rh</span> <span m="1489130">factor</span> <span m="1489920">and</span> <span m="1490160">then</span> <span m="1490300">1/4</span> <span m="1490980">of</span> <span m="1491060">all</span> <span m="1491215">of</span> <span m="1491370">those</span> <span m="1492190">have</span> <span m="1492400">this</span> <span m="1492550">marker,</span> <span m="1493320">that's</span> <span m="1493490">1</span> <span m="1493625">in</span> <span m="1493760">200.</span> <span m="1497430">Now,</span> <span m="1497680">it's</span> <span m="1497790">important</span> <span m="1498270">because</span> <span m="1499550">OJ's</span> <span m="1500070">blood</span> <span m="1500810">and</span> <span m="1500980">the</span> <span m="1501040">blood</span> <span m="1501670">at</span> <span m="1501880">the</span> <span m="1501960">crime</span> <span m="1502330">scene</span> <span m="1503630">both</span> <span m="1504670">matched</span> <span m="1505010">all</span> <span m="1505190">three.</span> <span m="1507310">So</span> <span m="1507520">the</span> <span m="1507640">implication,</span> <span m="1508330">of</span> <span m="1508430">course,</span> <span m="1508590">is</span> <span m="1508720">that</span> <span m="1508840">OJ</span> <span m="1508980">is</span> <span m="1509270">looking</span> <span m="1509470">like</span> <span m="1509710">the</span> <span m="1509780">guy</span> <span m="1510000">who</span> <span m="1510150">did</span> <span m="1510350">it.</span> <span m="1511620">And</span> <span m="1511780">the</span> <span m="1511860">question</span> <span m="1512260">was,</span> <span m="1512640">well,</span> <span m="1513640">is</span> <span m="1513900">the</span> <span m="1514010">1</span> <span m="1514195">in</span> <span m="1514380">200</span> <span m="1514980">really</span> <span m="1515250">true?</span> <span m="1516670">We</span> <span m="1516840">can</span> <span m="1516970">sample</span> <span m="1517800">these</span> <span m="1518080">three</span> <span m="1518290">in</span> <span m="1518335">the</span> <span m="1518380">populations</span> <span m="1519060">and</span> <span m="1519220">see</span> <span m="1519520">they're</span> <span m="1519620">true,</span> <span m="1519720">but</span> <span m="1520350">is</span> <span m="1520570">1</span> <span m="1520860">in</span> <span m="1520970">200</span> <span m="1521450">really</span> <span m="1521700">true?</span> </p>
<p><span m="1523850">Now,</span> <span m="1524560">it</span> <span m="1524710">would</span> <span m="1525030">be</span> <span m="1525540">if,</span> <span m="1525810">in</span> <span m="1525950">fact,</span> <span m="1526540">we</span> <span m="1526700">verified</span> <span m="1527660">that</span> <span m="1528090">1/5</span> <span m="1528540">of</span> <span m="1528640">the</span> <span m="1528710">type</span> <span m="1528930">O</span> <span m="1529150">people</span> <span m="1529490">have</span> <span m="1529700">positive</span> <span m="1531360">and</span> <span m="1531520">1/4</span> <span m="1532120">of</span> <span m="1532250">the</span> <span m="1532320">O</span> <span m="1532390">positive</span> <span m="1533140">people</span> <span m="1533480">have</span> <span m="1533870">the</span> <span m="1533980">XYZ</span> <span m="1534530">marker.</span> <span m="1536500">But</span> <span m="1537590">well,</span> <span m="1537820">we</span> <span m="1537940">don't</span> <span m="1538500">necessarily</span> <span m="1539160">know</span> <span m="1539480">that</span> <span m="1539860">unless</span> <span m="1540120">we</span> <span m="1540230">go</span> <span m="1540420">figure</span> <span m="1541110">that</span> <span m="1541350">out.</span> <span m="1542820">If</span> <span m="1542950">you</span> <span m="1543110">assume</span> <span m="1543650">they're</span> <span m="1543840">independent,</span> <span m="1545030">then</span> <span m="1545190">it</span> <span m="1545240">would</span> <span m="1545370">be</span> <span m="1545500">true.</span> <span m="1546280">The</span> <span m="1546380">product</span> <span m="1546820">rule</span> <span m="1546990">will</span> <span m="1547160">tell</span> <span m="1547390">us</span> <span m="1547520">that</span> <span m="1547690">if</span> <span m="1547770">you</span> <span m="1547900">assume</span> <span m="1548370">they're</span> <span m="1548530">independent.</span> </p>
<p><span m="1550280">So</span> <span m="1550740">during</span> <span m="1550970">the</span> <span m="1551080">trial,</span> <span m="1552090">a</span> <span m="1552190">special</span> <span m="1552640">math</span> <span m="1553320">defense</span> <span m="1553760">counsel</span> <span m="1554180">showed</span> <span m="1554460">up,</span> <span m="1555310">not</span> <span m="1555540">part</span> <span m="1555720">of</span> <span m="1555760">the</span> <span m="1555820">normal</span> <span m="1556180">defense</span> <span m="1556590">team,</span> <span m="1556790">but</span> <span m="1556860">he</span> <span m="1556920">was</span> <span m="1557070">brought</span> <span m="1557360">in</span> <span m="1558280">as</span> <span m="1558470">a</span> <span m="1558640">mathematician</span> <span m="1559005">and</span> <span m="1559370">lawyer</span> <span m="1560480">and</span> <span m="1560670">he</span> <span m="1561890">crosses</span> <span m="1562580">the</span> <span m="1562790">police</span> <span m="1563150">guy</span> <span m="1563300">on</span> <span m="1563610">the</span> <span m="1563920">stand.</span> <span m="1564570">And</span> <span m="1564790">he</span> <span m="1564990">asked</span> <span m="1565350">the</span> <span m="1565500">police</span> <span m="1565910">guy,</span> <span m="1566280">the</span> <span m="1566420">lab</span> <span m="1566700">guy</span> <span m="1567480">if</span> <span m="1567650">it</span> <span m="1567740">is</span> <span m="1567920">known</span> <span m="1568520">that</span> <span m="1568650">these</span> <span m="1568900">three</span> <span m="1569120">factors</span> <span m="1569840">are</span> <span m="1570470">independent.</span> <span m="1573350">Well,</span> <span m="1573670">the</span> <span m="1573760">poor</span> <span m="1574370">police</span> <span m="1574670">lab</span> <span m="1574930">guy</span> <span m="1575230">never</span> <span m="1575430">heard</span> <span m="1575640">the</span> <span m="1575720">word</span> <span m="1575900">independent</span> <span m="1576370">before,</span> <span m="1576790">didn't</span> <span m="1576870">know</span> <span m="1577010">what</span> <span m="1577180">it</span> <span m="1577260">meant</span> <span m="1578370">and</span> <span m="1579080">the</span> <span m="1579170">defense</span> <span m="1579600">counsel</span> <span m="1579990">proceeded</span> <span m="1580380">to</span> <span m="1580450">crucify</span> <span m="1580990">him</span> <span m="1581420">on</span> <span m="1581610">the</span> <span m="1581700">stand.</span> <span m="1582680">And</span> <span m="1582950">then</span> <span m="1583180">in</span> <span m="1583440">the</span> <span m="1583450">end,</span> <span m="1583460">all</span> <span m="1583470">he</span> <span m="1583480">could</span> <span m="1583610">say</span> <span m="1583725">was,</span> <span m="1583840">look,</span> <span m="1584310">we</span> <span m="1584440">just</span> <span m="1584650">get</span> <span m="1584840">these</span> <span m="1585040">things</span> <span m="1585170">and</span> <span m="1585300">we</span> <span m="1585440">multiply</span> <span m="1585740">them.</span> <span m="1586210">That's</span> <span m="1586430">what</span> <span m="1586500">we're</span> <span m="1586620">supposed</span> <span m="1586900">to</span> <span m="1586990">do.</span> </p>
<p><span m="1589500">It</span> <span m="1589905">was</span> <span m="1590310">a</span> <span m="1590360">little</span> <span m="1590500">scary.</span> <span m="1590950">The</span> <span m="1591030">actual</span> <span m="1591450">transcript--</span> <span m="1592480">you</span> <span m="1592560">can</span> <span m="1592690">still</span> <span m="1592880">get</span> <span m="1593050">it--</span> <span m="1593120">is</span> <span m="1593190">a</span> <span m="1593240">little</span> <span m="1593640">scary.</span> <span m="1594620">The</span> <span m="1594730">same</span> <span m="1595060">problem</span> <span m="1595450">arises</span> <span m="1595950">today</span> <span m="1596260">with</span> <span m="1596410">DNA</span> <span m="1596810">testing.</span> <span m="1597990">Only</span> <span m="1598280">there,</span> <span m="1599250">you've</span> <span m="1599380">got</span> <span m="1599820">lots</span> <span m="1600200">of</span> <span m="1600300">these</span> <span m="1600560">things</span> <span m="1601880">and</span> <span m="1602090">you</span> <span m="1602180">multiply</span> <span m="1602910">them</span> <span m="1603020">all</span> <span m="1603200">together</span> <span m="1603720">and</span> <span m="1603820">you</span> <span m="1603920">get</span> <span m="1603980">probabilities</span> <span m="1604620">like</span> <span m="1604870">one</span> <span m="1605210">in</span> <span m="1605880">many</span> <span m="1606150">billion</span> <span m="1607190">probability</span> <span m="1607790">of</span> <span m="1607870">a</span> <span m="1607920">match.</span> </p>
<p><span m="1610260">Now,</span> <span m="1610950">there's</span> <span m="1611140">probably</span> <span m="1611530">a</span> <span m="1611580">higher</span> <span m="1611870">level</span> <span m="1612140">of</span> <span m="1612200">science</span> <span m="1612620">going</span> <span m="1612840">on</span> <span m="1613410">with</span> <span m="1613520">DNA</span> <span m="1613900">testing,</span> <span m="1615430">but</span> <span m="1615560">it's</span> <span m="1615690">even</span> <span m="1615900">harder</span> <span m="1616370">to</span> <span m="1616610">really</span> <span m="1617080">establish</span> <span m="1617650">independence.</span> <span m="1619440">If</span> <span m="1619530">you</span> <span m="1619680">assume</span> <span m="1620200">it,</span> <span m="1620620">fine.</span> <span m="1621560">The</span> <span m="1621650">math</span> <span m="1621910">works</span> <span m="1622180">out</span> <span m="1622320">great.</span> <span m="1622690">You</span> <span m="1622790">just</span> <span m="1622990">multiply</span> <span m="1623220">them</span> <span m="1623450">together.</span> <span m="1624280">But</span> <span m="1624460">how</span> <span m="1624650">do</span> <span m="1624710">you</span> <span m="1624820">know</span> <span m="1625020">it's</span> <span m="1625150">really</span> <span m="1625470">true?</span> <span m="1626840">How</span> <span m="1627020">do</span> <span m="1627080">you</span> <span m="1627540">know</span> <span m="1628000">that</span> <span m="1628230">maybe</span> <span m="1628460">a</span> <span m="1628520">lot</span> <span m="1628900">of</span> <span m="1628960">people</span> <span m="1629210">that</span> <span m="1629310">have</span> <span m="1629500">those</span> <span m="1629840">four</span> <span m="1630330">markers</span> <span m="1630790">and</span> <span m="1630890">DNA</span> <span m="1631310">don't</span> <span m="1631590">happen</span> <span m="1631910">to</span> <span m="1631980">just</span> <span m="1632250">have</span> <span m="1632400">the</span> <span m="1632500">fifth</span> <span m="1632880">also,</span> <span m="1634120">but</span> <span m="1634300">it</span> <span m="1634470">really</span> <span m="1634770">is</span> <span m="1635160">totally</span> <span m="1635770">unrelated.</span> </p>
<p><span m="1637160">And</span> <span m="1637310">to</span> <span m="1637390">know</span> <span m="1637610">that</span> <span m="1637820">for</span> <span m="1637920">sure,</span> <span m="1638900">you</span> <span m="1639060">got</span> <span m="1639210">to</span> <span m="1639280">test</span> <span m="1639630">hundreds</span> <span m="1640060">of</span> <span m="1640120">millions</span> <span m="1640460">of</span> <span m="1640540">people,</span> <span m="1641920">which</span> <span m="1642290">we</span> <span m="1642450">really</span> <span m="1642680">haven't</span> <span m="1643060">done</span> <span m="1643270">yet,</span> <span m="1643840">and</span> <span m="1644020">not</span> <span m="1644190">just</span> <span m="1644380">a</span> <span m="1644430">few</span> <span m="1644620">guys</span> <span m="1645070">in</span> <span m="1645150">Detroit</span> <span m="1646320">to</span> <span m="1646430">be</span> <span m="1646510">able</span> <span m="1646610">to</span> <span m="1646670">conclude</span> <span m="1648330">independence</span> <span m="1648930">of</span> <span m="1649030">1</span> <span m="1649280">in</span> <span m="1649315">a</span> <span m="1649350">billion</span> <span m="1649670">probabilities.</span> </p>
<p><span m="1651800">So</span> <span m="1651990">for</span> <span m="1652180">us,</span> <span m="1652330">this</span> <span m="1652470">is</span> <span m="1652580">a</span> <span m="1652630">lot</span> <span m="1652860">easier.</span> <span m="1653160">In</span> <span m="1653220">the</span> <span m="1653300">classroom,</span> <span m="1653860">we</span> <span m="1654000">assume</span> <span m="1654390">independence</span> <span m="1654670">and</span> <span m="1654950">we'll</span> <span m="1655100">keep</span> <span m="1655320">doing</span> <span m="1655520">that</span> <span m="1655690">left</span> <span m="1655950">and</span> <span m="1656030">right,</span> <span m="1657550">but</span> <span m="1657680">it</span> <span m="1657770">doesn't</span> <span m="1658360">mean</span> <span m="1658630">it's</span> <span m="1658800">true</span> <span m="1660210">in</span> <span m="1660360">reality.</span> <span m="1660620">In</span> <span m="1660880">fact,</span> <span m="1662070">in</span> <span m="1662190">the</span> <span m="1662370">last</span> <span m="1662700">week</span> <span m="1662810">of</span> <span m="1662870">class.</span> <span m="1663380">We'll</span> <span m="1663590">talk</span> <span m="1663850">about</span> <span m="1664130">how</span> <span m="1664430">false</span> <span m="1665000">assumption</span> <span m="1665410">of</span> <span m="1665490">independence</span> <span m="1666720">on</span> <span m="1666930">mortgage</span> <span m="1667370">failures</span> <span m="1668650">led</span> <span m="1668980">to</span> <span m="1669060">the</span> <span m="1669170">subprime</span> <span m="1669640">mortgage</span> <span m="1670000">disaster</span> <span m="1670490">in</span> <span m="1670560">the</span> <span m="1670640">recession.</span> <span m="1671780">It</span> <span m="1671920">was</span> <span m="1672040">all</span> <span m="1672220">because</span> <span m="1672560">of</span> <span m="1672740">some</span> <span m="1672790">mathematics</span> <span m="1673330">mistakes</span> <span m="1674180">that</span> <span m="1674440">people</span> <span m="1674710">made.</span> </p>
<p><span m="1677530">Now,</span> <span m="1677960">this</span> <span m="1678140">example</span> <span m="1678600">raises</span> <span m="1678930">the</span> <span m="1679010">question</span> <span m="1679460">of,</span> <span m="1679910">what</span> <span m="1680180">does</span> <span m="1680320">independence</span> <span m="1680920">mean</span> <span m="1681160">when</span> <span m="1681270">you</span> <span m="1681340">have</span> <span m="1681520">more</span> <span m="1681720">than</span> <span m="1681900">two</span> <span m="1682160">events?</span> <span m="1684570">We</span> <span m="1684750">defined</span> <span m="1685220">independence</span> <span m="1685670">when</span> <span m="1685750">there</span> <span m="1685820">is</span> <span m="1685890">two</span> <span m="1686030">events,</span> <span m="1686360">but</span> <span m="1686450">here</span> <span m="1686680">there's</span> <span m="1686980">three.</span> <span m="1688280">And</span> <span m="1688460">so</span> <span m="1688600">to</span> <span m="1688660">be</span> <span m="1688780">careful,</span> <span m="1689180">we</span> <span m="1689270">got</span> <span m="1689410">to</span> <span m="1689770">actually</span> <span m="1690080">define</span> <span m="1691450">dependence</span> <span m="1691990">among</span> <span m="1692300">more</span> <span m="1692680">than</span> <span m="1693160">two</span> <span m="1693370">events</span> <span m="1695360">and</span> <span m="1695430">in</span> <span m="1695490">this</span> <span m="1695690">case,</span> <span m="1696010">we</span> <span m="1696150">talk</span> <span m="1696480">about</span> <span m="1697690">the</span> <span m="1697830">events</span> <span m="1698200">as</span> <span m="1698320">being</span> <span m="1698570">mutually</span> <span m="1699250">independent.</span> <span m="1700990">So</span> <span m="1701140">let</span> <span m="1701250">me</span> <span m="1701360">define</span> <span m="1701820">that.</span> </p>
<p><span m="1716770">So</span> <span m="1716810">if</span> <span m="1716910">I've</span> <span m="1717030">got</span> <span m="1717230">events</span> <span m="1718450">A1,</span> <span m="1719890">A2,</span> <span m="1720874">up</span> <span m="1721366">to</span> <span m="1721860">An,</span> <span m="1724330">we</span> <span m="1724450">say</span> <span m="1724710">they</span> <span m="1724940">are</span> <span m="1725420">mutually</span> <span m="1726130">independent</span> <span m="1735670">if,</span> <span m="1736120">and</span> <span m="1736220">this</span> <span m="1736380">is</span> <span m="1736470">a</span> <span m="1736520">little</span> <span m="1736780">complicated</span> <span m="1737450">notation,</span> <span m="1739150">but</span> <span m="1739520">for</span> <span m="1739710">all</span> <span m="1740300">i</span> <span m="1742490">and</span> <span m="1742860">for</span> <span m="1743090">all</span> <span m="1743700">sets</span> <span m="1744780">j</span> <span m="1746480">that</span> <span m="1747040">are</span> <span m="1747340">subsets</span> <span m="1747840">of</span> <span m="1747930">the</span> <span m="1748070">events,</span> <span m="1750740">but</span> <span m="1750980">not</span> <span m="1751200">including</span> <span m="1751610">i,</span> <span m="1757500">then</span> <span m="1757770">the</span> <span m="1757880">probability</span> <span m="1759380">that</span> <span m="1759460">the</span> <span m="1759610">i-th</span> <span m="1759910">event</span> <span m="1760120">occurs</span> <span m="1761260">given</span> <span m="1762560">that</span> <span m="1763070">all</span> <span m="1763340">the</span> <span m="1763480">events</span> <span m="1763890">in</span> <span m="1763940">the</span> <span m="1764020">subset</span> <span m="1764490">occurred,</span> <span m="1768280">is</span> <span m="1768540">the</span> <span m="1768620">same</span> <span m="1769010">as</span> <span m="1769140">the</span> <span m="1769240">probability</span> <span m="1769770">of</span> <span m="1769840">the</span> <span m="1769940">i-th</span> <span m="1770210">event</span> <span m="1770480">occurring</span> <span m="1770870">by</span> <span m="1771050">itself.</span> <span m="1773280">Or</span> <span m="1773670">there's</span> <span m="1773880">a</span> <span m="1773940">special</span> <span m="1774280">case</span> <span m="1774660">where</span> <span m="1775080">the</span> <span m="1775220">chance</span> <span m="1775570">the</span> <span m="1775810">other</span> <span m="1776060">events</span> <span m="1776390">occur</span> <span m="1776680">is</span> <span m="1776830">0.</span> </p>
<p><span m="1786710">In</span> <span m="1786890">other</span> <span m="1787100">words,</span> <span m="1787650">a</span> <span m="1787750">collection</span> <span m="1788250">of</span> <span m="1788320">events</span> <span m="1788740">is</span> <span m="1788860">mutually</span> <span m="1789330">independent</span> <span m="1791020">if</span> <span m="1791430">any</span> <span m="1791740">knowledge</span> <span m="1792220">about</span> <span m="1792580">any</span> <span m="1793510">of</span> <span m="1793640">the</span> <span m="1793730">rest</span> <span m="1794020">of</span> <span m="1794060">the</span> <span m="1794190">events,</span> <span m="1795070">happening</span> <span m="1795520">or</span> <span m="1795570">not,</span> <span m="1795960">does</span> <span m="1796110">not</span> <span m="1796390">influence</span> <span m="1797730">the</span> <span m="1797830">event</span> <span m="1798110">you're</span> <span m="1798210">looking</span> <span m="1798470">at</span> <span m="1799030">for</span> <span m="1799190">each</span> <span m="1799410">of</span> <span m="1799480">those</span> <span m="1799670">events.</span> <span m="1800970">So</span> <span m="1801040">no</span> <span m="1801380">information</span> <span m="1801860">about</span> <span m="1802080">any</span> <span m="1802320">of</span> <span m="1802360">the</span> <span m="1802500">other</span> <span m="1802660">markers</span> <span m="1803080">the</span> <span m="1803190">blood</span> <span m="1803950">influences</span> <span m="1804630">the</span> <span m="1804760">i-th</span> <span m="1805045">marker</span> <span m="1805690">for</span> <span m="1805870">any</span> <span m="1806060">i.</span> <span m="1806980">The</span> <span m="1807080">probabilities</span> <span m="1807660">are</span> <span m="1807770">unchanged.</span> </p>
<p><span m="1809770">Now,</span> <span m="1809900">there's</span> <span m="1810080">an</span> <span m="1810150">equivalent</span> <span m="1810790">definitions</span> <span m="1811430">based</span> <span m="1811710">and</span> <span m="1811780">the</span> <span m="1811860">product</span> <span m="1812340">rule.</span> <span m="1814691">Let</span> <span m="1815160">me</span> <span m="1815300">show</span> <span m="1815440">you</span> <span m="1815580">that</span> <span m="1815840">version</span> <span m="1816040">because</span> <span m="1816240">that's</span> <span m="1816480">easier</span> <span m="1816860">to</span> <span m="1816940">work</span> <span m="1817240">with</span> <span m="1817400">usually.</span> <span m="1832850">This</span> <span m="1833040">is</span> <span m="1833170">the</span> <span m="1833250">product</span> <span m="1833750">rule</span> <span m="1834000">form</span> <span m="1843130">and</span> <span m="1843740">it</span> <span m="1843880">says</span> <span m="1844360">that</span> <span m="1844600">A1,</span> <span m="1845280">A2,</span> <span m="1846992">up</span> <span m="1847420">to</span> <span m="1847850">An</span> <span m="1849100">are</span> <span m="1850090">mutually</span> <span m="1850550">independent</span> <span m="1857900">if</span> <span m="1858480">for</span> <span m="1858810">any</span> <span m="1859110">subset</span> <span m="1859740">of</span> <span m="1859840">the</span> <span m="1859960">events</span> <span m="1867560">the</span> <span m="1867910">probability</span> <span m="1869210">of</span> <span m="1869730">each</span> <span m="1869960">of</span> <span m="1870030">those</span> <span m="1870280">events</span> <span m="1870660">in</span> <span m="1870730">the</span> <span m="1870800">subset</span> <span m="1871280">happening,</span> <span m="1871880">all</span> <span m="1872170">them</span> <span m="1872310">happening,</span> <span m="1876140">is</span> <span m="1876300">simply</span> <span m="1876750">the</span> <span m="1876860">product</span> <span m="1878500">of</span> <span m="1878700">their</span> <span m="1878890">individual</span> <span m="1879400">probabilities.</span> </p>
<p><span m="1886810">So</span> <span m="1887330">independence</span> <span m="1889260">means</span> <span m="1889810">that</span> <span m="1890740">if</span> <span m="1890800">you</span> <span m="1890910">want</span> <span m="1891060">the</span> <span m="1891160">probability</span> <span m="1891650">of</span> <span m="1891710">a</span> <span m="1891790">bunch</span> <span m="1892060">of</span> <span m="1892150">events</span> <span m="1892440">occurring,</span> <span m="1893430">just</span> <span m="1893940">multiply</span> <span m="1894480">them</span> <span m="1894630">out</span> <span m="1894740">individually.</span> <span m="1896060">And</span> <span m="1896190">that</span> <span m="1896430">follows</span> <span m="1896860">for</span> <span m="1897030">independence</span> <span m="1897370">or</span> <span m="1897710">it</span> <span m="1897910">could</span> <span m="1898050">be</span> <span m="1898310">the</span> <span m="1898440">definition</span> <span m="1898990">of</span> <span m="1899060">independence,</span> <span m="1899600">depending</span> <span m="1899900">on</span> <span m="1900000">how</span> <span m="1900040">you</span> <span m="1900160">want</span> <span m="1900330">to</span> <span m="1900400">do</span> <span m="1900590">it.</span> <span m="1901080">So</span> <span m="1901250">either</span> <span m="1901405">of</span> <span m="1901560">these</span> <span m="1901800">are</span> <span m="1901860">good</span> <span m="1902040">enough</span> <span m="1902290">for</span> <span m="1902390">you</span> <span m="1902460">to</span> <span m="1902530">use</span> <span m="1902890">as</span> <span m="1903330">a</span> <span m="1903400">definition</span> <span m="1903990">or</span> <span m="1904180">a</span> <span m="1904270">result</span> <span m="1905090">for</span> <span m="1905240">independence.</span> <span m="1907200">And</span> <span m="1907350">so</span> <span m="1907450">the</span> <span m="1907540">blood</span> <span m="1907810">guy,</span> <span m="1908000">of</span> <span m="1908080">course,</span> <span m="1908440">is</span> <span m="1908540">just</span> <span m="1908920">multiplying</span> <span m="1909220">them</span> <span m="1909520">out</span> <span m="1910030">because</span> <span m="1911020">they're</span> <span m="1911180">assumed</span> <span m="1911490">to</span> <span m="1911550">be</span> <span m="1911670">independent,</span> <span m="1912390">so</span> <span m="1912600">it's</span> <span m="1912730">OK</span> <span m="1913000">that</span> <span m="1913240">way.</span> </p>
<p><span m="1915180">Let's</span> <span m="1915500">do</span> <span m="1915600">an</span> <span m="1915680">example.</span> <span m="1928920">So</span> <span m="1929340">for</span> <span m="1929520">example,</span> <span m="1929900">say</span> <span m="1930100">we</span> <span m="1930210">have</span> <span m="1930380">three</span> <span m="1930630">events.</span> <span m="1935160">A1,</span> <span m="1936440">A2,</span> <span m="1936955">and</span> <span m="1937280">A3</span> <span m="1939570">are</span> <span m="1939850">mutually</span> <span m="1940310">independent</span> <span m="1945430">if,</span> <span m="1946190">these</span> <span m="1946450">are</span> <span m="1946490">the</span> <span m="1946580">things</span> <span m="1946800">you</span> <span m="1946900">have</span> <span m="1947020">to</span> <span m="1947120">check,</span> <span m="1948450">probability</span> <span m="1949220">A1</span> <span m="1950970">and</span> <span m="1951130">A2</span> <span m="1952810">is</span> <span m="1953140">just</span> <span m="1953370">the</span> <span m="1954820">probability</span> <span m="1955190">of</span> <span m="1955340">A1</span> <span m="1955600">times</span> <span m="1955770">the</span> <span m="1955940">probability</span> <span m="1956340">of</span> <span m="1956440">A2.</span> <span m="1959160">Then</span> <span m="1959330">you'd</span> <span m="1959550">check</span> <span m="1961406">that</span> <span m="1961820">the</span> <span m="1962160">probability</span> <span m="1962600">of</span> <span m="1962800">A1</span> <span m="1963050">and</span> <span m="1963140">A3</span> <span m="1964900">is</span> <span m="1965090">the</span> <span m="1965180">product</span> <span m="1965610">of</span> <span m="1965690">their</span> <span m="1965880">probabilities,</span> <span m="1967490">A1</span> <span m="1967750">and</span> <span m="1967890">A3.</span> <span m="1974170">And</span> <span m="1974490">you'd</span> <span m="1974640">check</span> <span m="1975390">the</span> <span m="1975450">probability</span> <span m="1976180">of</span> <span m="1976950">A2</span> <span m="1977250">and</span> <span m="1977550">A3</span> <span m="1978950">is</span> <span m="1979260">the</span> <span m="1979350">product</span> <span m="1979810">of</span> <span m="1979890">their</span> <span m="1980090">probabilities.</span> </p>
<p><span m="1986160">And</span> <span m="1986400">there's</span> <span m="1986550">one</span> <span m="1986760">more</span> <span m="1986930">thing</span> <span m="1987130">to</span> <span m="1987220">check.</span> <span m="1987550">What's</span> <span m="1987740">that?</span> <span m="1990290">All</span> <span m="1990650">of</span> <span m="1990700">them.</span> <span m="1992120">The</span> <span m="1992250">probability</span> <span m="1993790">of</span> <span m="1993860">all</span> <span m="1994160">of</span> <span m="1994260">them</span> <span m="2000230">is</span> <span m="2000440">the</span> <span m="2000550">product</span> <span m="2001420">of</span> <span m="2001590">each</span> <span m="2001810">of</span> <span m="2002010">them</span> <span m="2003090">together</span> <span m="2003390">here.</span> <span m="2007940">So</span> <span m="2008100">if</span> <span m="2008170">you</span> <span m="2008280">want</span> <span m="2008470">to</span> <span m="2008520">show</span> <span m="2008950">the</span> <span m="2009090">three</span> <span m="2009470">events</span> <span m="2009820">are</span> <span m="2009870">mutually</span> <span m="2010220">independent,</span> <span m="2011050">these</span> <span m="2011310">are</span> <span m="2011340">the</span> <span m="2011410">four</span> <span m="2011640">things</span> <span m="2011890">you</span> <span m="2012030">check.</span> <span m="2013410">That's</span> <span m="2013610">one</span> <span m="2013790">way</span> <span m="2013880">to</span> <span m="2013990">do</span> <span m="2014190">it,</span> <span m="2016170">which</span> <span m="2016400">is</span> <span m="2016700">the</span> <span m="2016810">case</span> <span m="2017090">of</span> <span m="2017145">the</span> <span m="2017200">blood</span> <span m="2017460">typing</span> <span m="2017850">in</span> <span m="2017895">the</span> <span m="2017940">situation.</span> </p>
<p><span m="2020816">All</span> <span m="2021260">right.</span> <span m="2021530">Let's</span> <span m="2021730">do</span> <span m="2023210">an</span> <span m="2023280">example.</span> <span m="2029470">Well,</span> <span m="2029730">for</span> <span m="2029840">example,</span> <span m="2030700">if</span> <span m="2030780">I</span> <span m="2030850">flip</span> <span m="2031280">three</span> <span m="2032630">unbiased,</span> <span m="2033510">mutually</span> <span m="2033890">independent</span> <span m="2034230">coins.</span> <span m="2035290">The</span> <span m="2035790">probability</span> <span m="2036310">of</span> <span m="2036430">two</span> <span m="2036690">of</span> <span m="2036740">them</span> <span m="2036910">being</span> <span m="2037160">heads</span> <span m="2037540">is</span> <span m="2037640">1/4.</span> <span m="2038170">The</span> <span m="2038270">probability</span> <span m="2038680">of</span> <span m="2038840">three</span> <span m="2039000">being</span> <span m="2039240">heads</span> <span m="2039590">is</span> <span m="2040300">1/8</span> <span m="2041010">and</span> <span m="2041140">so</span> <span m="2041330">forth.</span> <span m="2045010">Let's</span> <span m="2045310">do</span> <span m="2045960">a</span> <span m="2046060">trickier</span> <span m="2046560">example.</span> <span m="2047750">This</span> <span m="2047980">is</span> <span m="2048120">a</span> <span m="2048219">question</span> <span m="2048710">that</span> <span m="2048730">was</span> <span m="2048889">on</span> <span m="2049000">the</span> <span m="2049090">final</span> <span m="2049420">exam</span> <span m="2049900">a</span> <span m="2049949">few</span> <span m="2050139">years</span> <span m="2050400">ago</span> <span m="2052000">and</span> <span m="2053389">a</span> <span m="2053460">lot</span> <span m="2053750">of</span> <span m="2053790">the</span> <span m="2053880">class</span> <span m="2054195">missed</span> <span m="2054510">it.</span> <span m="2055340">So</span> <span m="2055590">now</span> <span m="2055840">we'll</span> <span m="2055980">do</span> <span m="2056179">it</span> <span m="2056239">here.</span> </p>
<p><span m="2061610">Say</span> <span m="2061775">I</span> <span m="2061940">flip</span> <span m="2063469">three</span> <span m="2064690">fair,</span> <span m="2067050">mutually</span> <span m="2067580">independent</span> <span m="2072020">coins</span> <span m="2075449">and</span> <span m="2075710">my</span> <span m="2076110">events</span> <span m="2076719">are</span> <span m="2076780">going</span> <span m="2076880">to</span> <span m="2076960">be</span> <span m="2078420">A1</span> <span m="2079300">is</span> <span m="2079500">the</span> <span m="2079670">event</span> <span m="2082360">coin</span> <span m="2082800">1</span> <span m="2084480">matches</span> <span m="2084889">coin</span> <span m="2085150">2.</span> <span m="2091449">The</span> <span m="2091570">second</span> <span m="2091989">event,</span> <span m="2092320">A2,</span> <span m="2094050">is</span> <span m="2094230">the</span> <span m="2094360">event</span> <span m="2094730">that</span> <span m="2094850">coin</span> <span m="2095429">2</span> <span m="2096080">matches</span> <span m="2097460">coin</span> <span m="2097950">3.</span> <span m="2099420">And</span> <span m="2099910">the</span> <span m="2100000">third</span> <span m="2100310">event,</span> <span m="2101540">A3,</span> <span m="2103030">is</span> <span m="2103190">the</span> <span m="2103320">event</span> <span m="2103750">that</span> <span m="2104150">coin</span> <span m="2104450">3</span> <span m="2105570">matches</span> <span m="2106030">coin</span> <span m="2106420">1.</span> </p>
<p><span m="2109980">And</span> <span m="2110280">the</span> <span m="2110350">question</span> <span m="2110820">was,</span> <span m="2112290">are</span> <span m="2112500">these</span> <span m="2113120">three</span> <span m="2113520">events</span> <span m="2115180">mutually</span> <span m="2115650">independent?</span> <span m="2118040">Prove</span> <span m="2118280">your</span> <span m="2118440">answer.</span> <span m="2121040">Let's</span> <span m="2121240">try</span> <span m="2121430">to</span> <span m="2121610">figure</span> <span m="2121850">that</span> <span m="2122080">out.</span> <span m="2131960">The</span> <span m="2132060">coins,</span> <span m="2132490">of</span> <span m="2132570">course,</span> <span m="2132930">are</span> <span m="2132980">mutually</span> <span m="2133320">independent,</span> <span m="2133810">but</span> <span m="2133820">what</span> <span m="2133980">about</span> <span m="2134270">these</span> <span m="2134520">events?</span> <span m="2136150">So</span> <span m="2136560">let's</span> <span m="2137050">start</span> <span m="2137320">doing</span> <span m="2137530">it.</span> <span m="2137640">What's</span> <span m="2137860">the</span> <span m="2137960">probability</span> <span m="2140140">one</span> <span m="2140520">of</span> <span m="2140570">the</span> <span m="2140720">events</span> <span m="2141110">occurring?</span> <span m="2144510">Well,</span> <span m="2146340">you</span> <span m="2146470">got</span> <span m="2146650">to</span> <span m="2146690">get</span> <span m="2147330">the</span> <span m="2147440">two</span> <span m="2147660">coins</span> <span m="2148100">at</span> <span m="2148170">hand</span> <span m="2148570">to</span> <span m="2148650">match,</span> <span m="2149840">so</span> <span m="2149990">that's</span> <span m="2150240">the</span> <span m="2150330">probability</span> <span m="2150930">of</span> <span m="2151000">a</span> <span m="2151040">heads,</span> <span m="2151370">heads</span> <span m="2153160">plus</span> <span m="2153425">the</span> <span m="2153690">probability</span> <span m="2154130">of</span> <span m="2154180">a</span> <span m="2154260">tails,</span> <span m="2154660">tails.</span> <span m="2156710">That's</span> <span m="2157310">1/4</span> <span m="2158220">plus</span> <span m="2158670">1/4</span> <span m="2159910">equals</span> <span m="2160260">1/2.</span> </p>
<p><span m="2165920">Now,</span> <span m="2166220">the</span> <span m="2166330">probability</span> <span m="2168920">of</span> <span m="2169250">Ai</span> <span m="2169830">and</span> <span m="2170340">Aj,</span> <span m="2171480">i</span> <span m="2171670">and</span> <span m="2171816">j</span> <span m="2171963">are</span> <span m="2172110">1</span> <span m="2172350">to</span> <span m="2172430">3,</span> <span m="2173280">they're</span> <span m="2173570">different,</span> <span m="2175780">but</span> <span m="2175930">what</span> <span m="2176090">is</span> <span m="2176500">a</span> <span m="2176560">way</span> <span m="2176740">of</span> <span m="2176820">characterizing</span> <span m="2177570">that</span> <span m="2177860">case?</span> <span m="2180120">Say</span> <span m="2180510">event</span> <span m="2180690">1</span> <span m="2180920">occurred</span> <span m="2181105">and</span> <span m="2181290">event</span> <span m="2181450">2</span> <span m="2181610">occurred,</span> <span m="2182920">how</span> <span m="2183130">would</span> <span m="2183190">I</span> <span m="2183250">characterize</span> <span m="2183510">that?</span> <span m="2188330">Yeah.</span> </p>
<p><span m="2188610">AUDIENCE: All</span> <span m="2188703">the</span> <span m="2188796">same.</span> </p>
<p><span m="2189782">TOM LEIGHTON:</span> <span m="2190230">All</span> <span m="2190700">of</span> <span m="2190806">them.</span> <span m="2190913">Yeah.</span> <span m="2191190">All</span> <span m="2191510">of</span> <span m="2191650">the</span> <span m="2191670">coins</span> <span m="2192010">are</span> <span m="2192140">the</span> <span m="2192220">same</span> <span m="2193300">because</span> <span m="2193470">if</span> <span m="2193560">A1</span> <span m="2193730">and</span> <span m="2194170">A2</span> <span m="2194340">occur,</span> <span m="2194410">I</span> <span m="2194650">know</span> <span m="2194810">1</span> <span m="2195080">matches</span> <span m="2195450">2</span> <span m="2195720">a</span> <span m="2195800">2</span> <span m="2195970">matches</span> <span m="2196340">3.</span> <span m="2197290">If</span> <span m="2197480">A1</span> <span m="2197950">and</span> <span m="2198060">A3</span> <span m="2198430">happen,</span> <span m="2198940">1</span> <span m="2199260">matches</span> <span m="2199730">2</span> <span m="2199920">and</span> <span m="2199990">1</span> <span m="2200230">matches</span> <span m="2200600">3,</span> <span m="2200910">so</span> <span m="2200965">they're</span> <span m="2201020">all</span> <span m="2201140">the</span> <span m="2201240">same</span> <span m="2201890">and</span> <span m="2201990">the</span> <span m="2202070">same</span> <span m="2202340">for</span> <span m="2202480">A2</span> <span m="2202900">and</span> <span m="2203000">A3.</span> <span m="2203830">If</span> <span m="2203950">2</span> <span m="2204140">matches</span> <span m="2204480">3</span> <span m="2204620">and</span> <span m="2204760">3</span> <span m="2204900">matches</span> <span m="2205220">1,</span> <span m="2205490">they're</span> <span m="2205620">all</span> <span m="2205800">the</span> <span m="2205900">same.</span> <span m="2206630">So</span> <span m="2206800">this</span> <span m="2207140">is</span> <span m="2207240">the</span> <span m="2207350">same</span> <span m="2207840">as</span> <span m="2208060">saying</span> <span m="2209080">all</span> <span m="2209380">three</span> <span m="2209600">coins</span> <span m="2209950">are</span> <span m="2210010">the</span> <span m="2210080">same.</span> <span m="2213890">It</span> <span m="2214020">could</span> <span m="2214120">all</span> <span m="2214220">be</span> <span m="2214350">heads</span> <span m="2214730">or</span> <span m="2214840">all</span> <span m="2214950">be</span> <span m="2215270">tails.</span> </p>
<p><span m="2217160">And</span> <span m="2217370">that's</span> <span m="2217610">an</span> <span m="2217730">8</span> <span m="2218650">plus</span> <span m="2219050">8,</span> <span m="2220320">which</span> <span m="2220480">is</span> <span m="2220580">1/4</span> <span m="2222290">and</span> <span m="2223050">that</span> <span m="2223280">means</span> <span m="2223750">equals</span> <span m="2226100">the</span> <span m="2226170">probability</span> <span m="2226660">of</span> <span m="2226730">Ai</span> <span m="2227900">times</span> <span m="2228230">the</span> <span m="2228310">probability</span> <span m="2229040">of</span> <span m="2229140">Aj,</span> <span m="2231520">which</span> <span m="2231730">is</span> <span m="2231780">what</span> <span m="2231950">I</span> <span m="2232030">need</span> <span m="2232710">for</span> <span m="2233160">independence.</span> <span m="2235850">And</span> <span m="2236040">then</span> <span m="2236210">they</span> <span m="2236360">said</span> <span m="2236780">they're</span> <span m="2236900">done.</span> <span m="2239070">They</span> <span m="2239230">are</span> <span m="2239440">independent,</span> <span m="2241230">the</span> <span m="2241330">three</span> <span m="2241570">events.</span> <span m="2243520">You</span> <span m="2243680">like</span> <span m="2243870">that</span> <span m="2244030">answer?</span> <span m="2245640">What's</span> <span m="2245850">missing?</span> </p>
<p><span m="2248260">The</span> <span m="2248360">last</span> <span m="2248700">case.</span> <span m="2249410">They</span> <span m="2249520">didn't</span> <span m="2249750">check</span> <span m="2250070">the</span> <span m="2250160">last</span> <span m="2250570">case</span> <span m="2251770">and</span> <span m="2251880">we</span> <span m="2251980">got</span> <span m="2252130">to</span> <span m="2252200">do</span> <span m="2252360">that</span> <span m="2252560">to</span> <span m="2252640">have</span> <span m="2252770">mutual</span> <span m="2253090">independence.</span> <span m="2253680">So</span> <span m="2253780">let's</span> <span m="2254010">look</span> <span m="2254180">at</span> <span m="2254260">that.</span> <span m="2255681">The</span> <span m="2256070">last</span> <span m="2256410">case</span> <span m="2256580">is</span> <span m="2256750">probability</span> <span m="2257190">A1</span> <span m="2257800">intersect</span> <span m="2258350">A2</span> <span m="2258940">intersect</span> <span m="2259600">A3.</span> <span m="2262060">What</span> <span m="2262260">is</span> <span m="2262400">the</span> <span m="2262490">probability</span> <span m="2263240">that</span> <span m="2264200">all</span> <span m="2264470">three</span> <span m="2264690">events</span> <span m="2265000">occur?</span> </p>
<p><span m="2269950">Well,</span> <span m="2272180">the</span> <span m="2272300">coins</span> <span m="2272630">all</span> <span m="2272780">have</span> <span m="2272950">to</span> <span m="2273040">match,</span> <span m="2273380">right?</span> <span m="2275560">If</span> <span m="2275710">all</span> <span m="2275880">the</span> <span m="2275960">coins</span> <span m="2276240">match,</span> <span m="2276620">all</span> <span m="2276930">three</span> <span m="2277090">events</span> <span m="2277380">occur,</span> <span m="2277730">right?</span> <span m="2279830">And</span> <span m="2279960">what's</span> <span m="2280130">the</span> <span m="2280220">probability</span> <span m="2280750">all</span> <span m="2280910">3</span> <span m="2281090">coins</span> <span m="2281420">match?</span> <span m="2284260">1/4,</span> <span m="2284900">just</span> <span m="2285190">the</span> <span m="2285270">same</span> <span m="2285520">as</span> <span m="2285610">this,</span> <span m="2285730">is</span> <span m="2285850">1/4.</span> <span m="2287770">Does</span> <span m="2287920">that</span> <span m="2288210">equal</span> <span m="2289380">probability</span> <span m="2289990">of</span> <span m="2290080">A1</span> <span m="2290870">times</span> <span m="2291520">the</span> <span m="2291600">probability</span> <span m="2292090">of</span> <span m="2292180">A2</span> <span m="2293070">times</span> <span m="2293810">the</span> <span m="2293890">probability</span> <span m="2295280">of</span> <span m="2295390">A3?</span> </p>
<p><span m="2300090">What's</span> <span m="2300320">that?</span> <span m="2302120">1/8.</span> <span m="2303680">This</span> <span m="2303880">is</span> <span m="2303990">1/8.</span> <span m="2305620">They</span> <span m="2305780">are</span> <span m="2305870">not</span> <span m="2306090">equal.</span> <span m="2309270">They</span> <span m="2309440">are</span> <span m="2309520">not</span> <span m="2309970">mutually</span> <span m="2310360">independent</span> <span m="2310830">events.</span> <span m="2312779">All</span> <span m="2313252">right?</span> <span m="2317520">Any</span> <span m="2317660">questions</span> <span m="2318040">about</span> <span m="2318260">that?</span> <span m="2319702">It</span> <span m="2320090">might</span> <span m="2320420">well</span> <span m="2320670">be</span> <span m="2320780">something</span> <span m="2321010">like</span> <span m="2321180">this</span> <span m="2321303">on</span> <span m="2321426">the</span> <span m="2321550">final</span> <span m="2321840">this</span> <span m="2322030">year,</span> <span m="2322810">a</span> <span m="2322960">good,</span> <span m="2323030">decent</span> <span m="2323450">chance.</span> </p>
<p><span m="2324960">So</span> <span m="2325150">if</span> <span m="2325250">you</span> <span m="2325540">start</span> <span m="2325790">going</span> <span m="2325940">along,</span> <span m="2326280">looks</span> <span m="2326590">like</span> <span m="2326770">they're</span> <span m="2326920">independent,</span> <span m="2327610">but</span> <span m="2327730">you</span> <span m="2327820">forget</span> <span m="2328050">to</span> <span m="2328120">check</span> <span m="2328360">that</span> <span m="2328520">last</span> <span m="2328770">case,</span> <span m="2329130">which</span> <span m="2329190">shows</span> <span m="2329550">they're</span> <span m="2329710">not</span> <span m="2330190">mutual</span> <span m="2330530">independent.</span> <span m="2332580">So</span> <span m="2332700">you've</span> <span m="2332760">got</span> <span m="2332880">to</span> <span m="2332950">check</span> <span m="2333290">for</span> <span m="2333450">all</span> <span m="2333630">pairs</span> <span m="2334240">and</span> <span m="2334610">all</span> <span m="2334890">subsets</span> <span m="2335410">of</span> <span m="2335490">events</span> <span m="2336320">for</span> <span m="2336440">mutual</span> <span m="2336790">independence.</span> <span m="2340040">Any</span> <span m="2340870">questions</span> <span m="2341300">about</span> <span m="2341540">that?</span> </p>
<p><span m="2346370">Now,</span> <span m="2346930">this</span> <span m="2347170">is</span> <span m="2347300">actually</span> <span m="2347730">an</span> <span m="2347820">interesting</span> <span m="2348250">example</span> <span m="2349960">because</span> <span m="2350450">in</span> <span m="2350560">this</span> <span m="2350800">case,</span> <span m="2352830">all</span> <span m="2353200">pairs</span> <span m="2353840">were</span> <span m="2354000">independent</span> <span m="2354690">and</span> <span m="2354860">when</span> <span m="2354960">that</span> <span m="2355160">happens,</span> <span m="2356290">we</span> <span m="2356560">give</span> <span m="2356720">that</span> <span m="2356870">a</span> <span m="2356920">special</span> <span m="2357340">name</span> <span m="2358490">and</span> <span m="2358650">it's</span> <span m="2358800">called</span> <span m="2359240">pairwise</span> <span m="2360100">independence,</span> <span m="2360820">not</span> <span m="2361190">too</span> <span m="2361340">surprising.</span> <span m="2362415">And</span> <span m="2362880">that</span> <span m="2363030">can</span> <span m="2363160">be</span> <span m="2363260">useful</span> <span m="2365910">because</span> <span m="2366060">there's</span> <span m="2366180">many</span> <span m="2366430">times</span> <span m="2366790">where</span> <span m="2366890">you</span> <span m="2366990">do</span> <span m="2367190">get</span> <span m="2367370">pairwise</span> <span m="2367840">independence,</span> <span m="2368340">but</span> <span m="2368470">not</span> <span m="2368750">mutual</span> <span m="2369130">independence.</span> <span m="2370840">So</span> <span m="2371140">let</span> <span m="2371182">me</span> <span m="2371225">give</span> <span m="2371267">you</span> <span m="2371310">that</span> <span m="2371520">definition.</span> </p>
<p><span m="2378760">So</span> <span m="2379010">a</span> <span m="2379070">collection</span> <span m="2379440">of</span> <span m="2379510">events</span> <span m="2381360">A1</span> <span m="2381850">through</span> <span m="2382010">An</span> <span m="2384460">are</span> <span m="2384820">said</span> <span m="2385020">to</span> <span m="2385090">be</span> <span m="2385310">pairwise</span> <span m="2389610">independent</span> <span m="2395610">if</span> <span m="2396690">for</span> <span m="2396950">all</span> <span m="2397380">i</span> <span m="2398100">and</span> <span m="2398290">j,</span> <span m="2399830">where</span> <span m="2400460">i</span> <span m="2400760">doesn't</span> <span m="2401120">equal</span> <span m="2401480">j,</span> <span m="2404980">Ai</span> <span m="2405960">and</span> <span m="2406240">Aj</span> <span m="2407860">are</span> <span m="2408070">independent.</span> <span m="2414230">Now,</span> <span m="2414430">as</span> <span m="2414510">we</span> <span m="2414610">saw</span> <span m="2415010">in</span> <span m="2415080">this</span> <span m="2415280">example,</span> <span m="2415840">in</span> <span m="2415900">this</span> <span m="2416060">example,</span> <span m="2417620">it</span> <span m="2417780">was</span> <span m="2418120">pairwise</span> <span m="2418700">independence</span> <span m="2420110">because</span> <span m="2421040">the</span> <span m="2421120">probability</span> <span m="2421660">of</span> <span m="2421730">Ai</span> <span m="2422130">and</span> <span m="2422440">Aj</span> <span m="2423670">equaled</span> <span m="2424040">the</span> <span m="2424120">probability</span> <span m="2424680">of</span> <span m="2424740">Ai</span> <span m="2425080">times</span> <span m="2425490">the</span> <span m="2425560">probably</span> <span m="2425940">of</span> <span m="2426030">Aj.</span> <span m="2426460">For</span> <span m="2426675">any</span> <span m="2426890">pair,</span> <span m="2427360">it</span> <span m="2427430">was</span> <span m="2427570">true.</span> <span m="2429130">But</span> <span m="2429310">it</span> <span m="2429420">doesn't</span> <span m="2429730">imply</span> <span m="2430140">mutual</span> <span m="2430530">independence.</span> <span m="2432130">So</span> <span m="2432330">pairwise</span> <span m="2434270">does</span> <span m="2434420">not</span> <span m="2434830">imply</span> <span m="2435720">mutual.</span> <span m="2438870">Mutual</span> <span m="2439340">would</span> <span m="2439560">imply</span> <span m="2439890">pairwise</span> <span m="2440570">because</span> <span m="2441090">it's</span> <span m="2441240">true</span> <span m="2441430">for</span> <span m="2441620">every</span> <span m="2441840">subset</span> <span m="2443090">of</span> <span m="2443230">events.</span> </p>
<p><span m="2446610">All</span> <span m="2446800">right.</span> <span m="2447050">So</span> <span m="2447150">let's</span> <span m="2447360">go</span> <span m="2447470">back</span> <span m="2447890">for</span> <span m="2448210">OJ</span> <span m="2450230">and</span> <span m="2450430">see</span> <span m="2450860">what</span> <span m="2451210">would</span> <span m="2451620">have</span> <span m="2452030">happened.</span> <span m="2453670">What</span> <span m="2453890">can</span> <span m="2454020">you</span> <span m="2454160">say</span> <span m="2454460">about</span> <span m="2454680">the</span> <span m="2454770">probability</span> <span m="2455450">of</span> <span m="2455530">a</span> <span m="2455600">blood</span> <span m="2455990">match</span> <span m="2456390">for</span> <span m="2456530">a</span> <span m="2456600">random</span> <span m="2456950">person</span> <span m="2458510">if</span> <span m="2458680">you</span> <span m="2459000">only</span> <span m="2459420">knew</span> <span m="2459870">that</span> <span m="2460110">these</span> <span m="2460420">factors</span> <span m="2460920">were</span> <span m="2461120">pairwise</span> <span m="2461950">independent?</span> <span m="2464910">Say</span> <span m="2465340">you</span> <span m="2465500">only</span> <span m="2465850">knew</span> <span m="2466040">that.</span> <span m="2466530">You</span> <span m="2466640">didn't</span> <span m="2466860">know</span> <span m="2467030">they</span> <span m="2467100">were</span> <span m="2467170">mutually</span> <span m="2467580">independent,</span> <span m="2468140">but</span> <span m="2468150">you</span> <span m="2468250">knew</span> <span m="2468540">they</span> <span m="2468690">were</span> <span m="2468850">pairwise</span> <span m="2469480">independent</span> <span m="2469735">in</span> <span m="2469990">the</span> <span m="2470060">population.</span> <span m="2475210">What's</span> <span m="2475390">the</span> <span m="2475480">best</span> <span m="2475780">you</span> <span m="2475860">can</span> <span m="2476030">say</span> <span m="2476360">about</span> <span m="2476490">the</span> <span m="2476620">probability</span> <span m="2477000">a</span> <span m="2477130">random</span> <span m="2477500">person</span> <span m="2477860">matches</span> <span m="2478260">that</span> <span m="2478450">blood</span> <span m="2478650">profile,</span> <span m="2480550">an</span> <span m="2480670">upper</span> <span m="2480850">bound</span> <span m="2481130">on</span> <span m="2481200">the</span> <span m="2481270">probability?</span> <span m="2483560">Yeah.</span> </p>
<p><span m="2483815">AUDIENCE:</span> <span m="2484070">1</span> <span m="2484423">in</span> <span m="2484599">50.</span> </p>
<p><span m="2484776">TOM LEIGHTON:</span> <span m="2485130">1</span> <span m="2485615">in</span> <span m="2485857">50.</span> <span m="2486100">Yeah.</span> <span m="2487050">So</span> <span m="2487210">what</span> <span m="2487370">you</span> <span m="2487470">can</span> <span m="2487590">say</span> <span m="2487900">is</span> <span m="2488050">1</span> <span m="2488450">in</span> <span m="2488650">50,</span> <span m="2488850">but</span> <span m="2489080">nothing</span> <span m="2489480">better.</span> <span m="2490810">So</span> <span m="2490950">let's</span> <span m="2491160">see</span> <span m="2491330">why</span> <span m="2491655">1</span> <span m="2491980">in</span> <span m="2492090">50</span> <span m="2492470">works.</span> <span m="2498490">So</span> <span m="2499390">let's</span> <span m="2499800">let</span> <span m="2500410">M1</span> <span m="2501000">be</span> <span m="2501150">the</span> <span m="2501280">event</span> <span m="2501640">you</span> <span m="2501740">match</span> <span m="2502130">here,</span> <span m="2503370">M2</span> <span m="2503830">be</span> <span m="2503980">the</span> <span m="2504100">event</span> <span m="2504390">you</span> <span m="2504470">match</span> <span m="2504770">their,</span> <span m="2504980">and</span> <span m="2505160">M3</span> <span m="2505490">be</span> <span m="2505620">the</span> <span m="2505740">event</span> <span m="2506020">you</span> <span m="2506100">match</span> <span m="2506410">that.</span> <span m="2508300">The</span> <span m="2508390">probability</span> <span m="2508950">you</span> <span m="2509040">match</span> <span m="2509390">all</span> <span m="2509580">three</span> <span m="2514770">is</span> <span m="2515000">upper</span> <span m="2515250">bounded</span> <span m="2515650">by</span> <span m="2515750">the</span> <span m="2515860">probability</span> <span m="2516490">you</span> <span m="2516600">match</span> <span m="2516930">the</span> <span m="2517030">first</span> <span m="2517340">two</span> <span m="2520470">because</span> <span m="2522490">matching</span> <span m="2522900">all</span> <span m="2523100">three</span> <span m="2523360">is</span> <span m="2523510">a</span> <span m="2523810">subset</span> <span m="2524650">of</span> <span m="2524820">this.</span> </p>
<p><span m="2528220">Pairwise</span> <span m="2528850">independence</span> <span m="2529490">means</span> <span m="2530470">that</span> <span m="2530640">this</span> <span m="2530850">is</span> <span m="2530960">true.</span> <span m="2531390">This</span> <span m="2531610">equals</span> <span m="2532740">the</span> <span m="2532790">probability</span> <span m="2533270">of</span> <span m="2533360">matching</span> <span m="2533690">the</span> <span m="2533780">first</span> <span m="2534210">times</span> <span m="2534275">the</span> <span m="2534340">probability</span> <span m="2534710">of</span> <span m="2535050">matching</span> <span m="2535085">the</span> <span m="2535120">second.</span> <span m="2536760">The</span> <span m="2537170">probability</span> <span m="2537360">of</span> <span m="2537550">matching</span> <span m="2537850">the</span> <span m="2537930">first</span> <span m="2538260">is</span> <span m="2538350">1/10,</span> <span m="2539670">probably</span> <span m="2540030">of</span> <span m="2540210">matching</span> <span m="2540390">the</span> <span m="2540470">second</span> <span m="2540850">is</span> <span m="2540980">1/5,</span> <span m="2542690">so</span> <span m="2542840">this</span> <span m="2543050">is</span> <span m="2543160">1/50.</span> <span m="2546700">And</span> <span m="2547150">you</span> <span m="2547290">picked</span> <span m="2548260">the</span> <span m="2548360">best</span> <span m="2548690">two.</span> <span m="2549880">You</span> <span m="2550100">could</span> <span m="2550290">have</span> <span m="2550410">picked</span> <span m="2550830">these</span> <span m="2551120">two</span> <span m="2552430">and</span> <span m="2552610">said</span> <span m="2552820">it</span> <span m="2552880">was</span> <span m="2553240">at</span> <span m="2553350">most</span> <span m="2553670">1/20</span> <span m="2554810">or</span> <span m="2555190">those</span> <span m="2555540">two</span> <span m="2555690">and</span> <span m="2555940">said</span> <span m="2555966">it's</span> <span m="2555993">at</span> <span m="2556020">most</span> <span m="2556380">1/40.</span> <span m="2558070">But</span> <span m="2558410">you</span> <span m="2558580">were</span> <span m="2559280">clever</span> <span m="2559540">and</span> <span m="2559800">said,</span> <span m="2559990">OK,</span> <span m="2560270">I'm</span> <span m="2560380">going</span> <span m="2560480">to</span> <span m="2560550">take</span> <span m="2560800">these</span> <span m="2561060">two</span> <span m="2561760">and</span> <span m="2561880">use</span> <span m="2562070">that</span> <span m="2562260">as</span> <span m="2562340">my</span> <span m="2562470">upper</span> <span m="2562630">bound,</span> <span m="2563030">which</span> <span m="2563090">is</span> <span m="2563170">1/50.</span> </p>
<p><span m="2564590">And</span> <span m="2564760">it</span> <span m="2564880">might</span> <span m="2565230">well</span> <span m="2565510">be</span> <span m="2567200">that</span> <span m="2567970">1</span> <span m="2568365">in</span> <span m="2568562">50</span> <span m="2568760">people</span> <span m="2569500">match</span> <span m="2569910">all</span> <span m="2570170">three.</span> <span m="2571450">That</span> <span m="2571630">can</span> <span m="2571750">well</span> <span m="2571950">be.</span> <span m="2574010">Because</span> <span m="2574410">maybe</span> <span m="2575250">whenever</span> <span m="2575750">you're</span> <span m="2575950">O</span> <span m="2576070">positive,</span> <span m="2576660">you</span> <span m="2576750">have</span> <span m="2577000">marker</span> <span m="2577380">XYZ.</span> <span m="2577910">That's</span> <span m="2578140">possible,</span> <span m="2579230">potentially,</span> <span m="2580220">unless</span> <span m="2580920">we</span> <span m="2581060">find</span> <span m="2581340">out</span> <span m="2581450">otherwise.</span> </p>
<p><span m="2585830">What</span> <span m="2586170">if</span> <span m="2586290">I</span> <span m="2586380">tell</span> <span m="2586650">you</span> <span m="2586960">can't</span> <span m="2587380">assume</span> <span m="2587720">any</span> <span m="2588020">independence</span> <span m="2588680">at</span> <span m="2588740">all?</span> <span m="2590180">What</span> <span m="2590420">can</span> <span m="2590530">you</span> <span m="2590630">say</span> <span m="2590870">about</span> <span m="2591070">the</span> <span m="2591160">probability</span> <span m="2591710">of</span> <span m="2591790">a</span> <span m="2591860">blood</span> <span m="2592140">match</span> <span m="2592440">here</span> <span m="2592890">for</span> <span m="2593020">a</span> <span m="2593090">random</span> <span m="2593380">person?</span> <span m="2594650">Yeah.</span> </p>
<p><span m="2594900">AUDIENCE:</span> <span m="2595400">1/10.</span> </p>
<p><span m="2595900">TOM LEIGHTON: What</span> <span m="2596020">is</span> <span m="2596150">it?</span> </p>
<p><span m="2596290">AUDIENCE:</span> <span m="2596780">1/10.</span> </p>
<p><span m="2597270">TOM LEIGHTON: 1/10.</span> <span m="2599500">Because</span> <span m="2599990">if</span> <span m="2600220">they</span> <span m="2600340">match</span> <span m="2600460">all</span> <span m="2600720">three,</span> <span m="2600975">they</span> <span m="2601230">match</span> <span m="2601500">this</span> <span m="2602110">and</span> <span m="2602210">that</span> <span m="2602350">probability</span> <span m="2603510">is</span> <span m="2603650">1/10,</span> <span m="2604140">so</span> <span m="2604220">it's</span> <span m="2604300">at</span> <span m="2604380">most</span> <span m="2604700">1/10.</span> <span m="2606510">And</span> <span m="2606710">it</span> <span m="2606790">could</span> <span m="2607070">be</span> <span m="2607280">that</span> <span m="2607380">everybody</span> <span m="2607760">who's</span> <span m="2607970">O</span> <span m="2608570">is</span> <span m="2608830">O</span> <span m="2609210">positive</span> <span m="2610120">and</span> <span m="2610260">has</span> <span m="2610570">XYZ.</span> <span m="2611810">So</span> <span m="2611990">unless</span> <span m="2612240">you</span> <span m="2612350">have</span> <span m="2612610">more</span> <span m="2613580">information,</span> <span m="2614770">that's</span> <span m="2614980">the</span> <span m="2615070">best</span> <span m="2615390">you</span> <span m="2615480">can</span> <span m="2615640">say.</span> <span m="2615870">It</span> <span m="2615990">might</span> <span m="2616180">well</span> <span m="2616360">be</span> <span m="2616540">that's</span> <span m="2618230">the</span> <span m="2618340">answer.</span> <span m="2621720">Any</span> <span m="2621860">questions</span> <span m="2623850">about</span> <span m="2624140">that?</span> </p>
<p><span m="2626120">So</span> <span m="2626260">the</span> <span m="2626390">assumptions</span> <span m="2626960">really</span> <span m="2627260">matter.</span> <span m="2627600">The</span> <span m="2627690">more</span> <span m="2628120">independence</span> <span m="2628950">you</span> <span m="2629070">assume,</span> <span m="2630200">the</span> <span m="2630300">better</span> <span m="2630620">bounds</span> <span m="2631120">and</span> <span m="2631200">the</span> <span m="2631260">probability</span> <span m="2631790">you</span> <span m="2631910">get</span> <span m="2632470">of</span> <span m="2632730">a</span> <span m="2632920">match.</span> <span m="2637106">It's</span> <span m="2637580">a</span> <span m="2637630">little</span> <span m="2637830">bit</span> <span m="2637930">unrelated</span> <span m="2638390">to</span> <span m="2638460">this,</span> <span m="2638660">but</span> <span m="2638740">there</span> <span m="2638850">was</span> <span m="2638940">another</span> <span m="2639530">mathematics</span> <span m="2640170">dispute</span> <span m="2640990">at</span> <span m="2641140">the</span> <span m="2641370">OJ</span> <span m="2641600">trial.</span> <span m="2643380">It</span> <span m="2643530">turned</span> <span m="2643850">out</span> <span m="2644030">the</span> <span m="2644110">that</span> <span m="2644450">OJ</span> <span m="2644620">had</span> <span m="2644790">been</span> <span m="2645170">beating</span> <span m="2645580">up</span> <span m="2645750">Nicole</span> <span m="2646450">on</span> <span m="2646630">a</span> <span m="2646690">fairly</span> <span m="2647010">regular</span> <span m="2647350">basis</span> <span m="2647750">and</span> <span m="2647820">there</span> <span m="2647895">were</span> <span m="2647970">police</span> <span m="2648320">records</span> <span m="2648710">because</span> <span m="2648880">after</span> <span m="2649120">he'd</span> <span m="2649250">beat</span> <span m="2649480">her</span> <span m="2649610">up,</span> <span m="2649910">she'd</span> <span m="2650020">go</span> <span m="2650270">in</span> <span m="2650430">and</span> <span m="2651340">complain</span> <span m="2651730">to</span> <span m="2651770">the</span> <span m="2651870">police.</span> </p>
<p><span m="2653250">And</span> <span m="2653470">the</span> <span m="2653540">prosecution</span> <span m="2654490">wanted</span> <span m="2654980">this</span> <span m="2655190">evidence</span> <span m="2655860">admitted</span> <span m="2656340">at</span> <span m="2656430">the</span> <span m="2656520">trial</span> <span m="2658410">because</span> <span m="2659420">if</span> <span m="2659580">the</span> <span m="2659660">guy</span> <span m="2659805">is</span> <span m="2659950">a</span> <span m="2660030">wife</span> <span m="2660420">beater,</span> <span m="2662410">it</span> <span m="2662520">makes</span> <span m="2662700">you</span> <span m="2662800">think</span> <span m="2663080">that</span> <span m="2663160">maybe</span> <span m="2663590">he</span> <span m="2663750">killed</span> <span m="2664050">her.</span> <span m="2665030">And</span> <span m="2665580">the</span> <span m="2665700">defense</span> <span m="2666180">lawyers</span> <span m="2666690">argued</span> <span m="2667070">against</span> <span m="2667690">admitting</span> <span m="2668220">that</span> <span m="2668410">evidence</span> <span m="2668810">because</span> <span m="2669090">it</span> <span m="2669150">wasn't</span> <span m="2669550">tied</span> <span m="2670090">to</span> <span m="2670980">the</span> <span m="2671120">actual</span> <span m="2671490">murder</span> <span m="2671900">scene</span> <span m="2672180">in</span> <span m="2672270">any</span> <span m="2672440">way</span> <span m="2673070">and</span> <span m="2673270">they</span> <span m="2673410">argued</span> <span m="2673760">it</span> <span m="2673810">would</span> <span m="2673980">be</span> <span m="2674110">prejudicial</span> <span m="2674790">to</span> <span m="2674860">the</span> <span m="2674980">jury</span> <span m="2675900">because,</span> <span m="2676440">of</span> <span m="2676530">course,</span> <span m="2676940">if</span> <span m="2677050">the</span> <span m="2677140">jury</span> <span m="2677500">hears</span> <span m="2677780">that</span> <span m="2677870">OJ</span> <span m="2678160">was</span> <span m="2678340">beating</span> <span m="2678660">her,</span> <span m="2679380">they</span> <span m="2679480">might</span> <span m="2680030">be</span> <span m="2680140">more</span> <span m="2680300">likely</span> <span m="2680570">to</span> <span m="2680670">include</span> <span m="2681000">to</span> <span m="2681170">convict</span> <span m="2681410">him</span> <span m="2681650">for</span> <span m="2681760">murdering</span> <span m="2682200">her.</span> </p>
<p><span m="2683690">Now,</span> <span m="2684850">they</span> <span m="2684960">got</span> <span m="2685140">the</span> <span m="2685210">math</span> <span m="2685490">council</span> <span m="2685890">again</span> <span m="2686620">to</span> <span m="2686750">argue</span> <span m="2687180">that</span> <span m="2688180">the</span> <span m="2688330">reason</span> <span m="2688680">you</span> <span m="2688780">shouldn't</span> <span m="2689250">admit</span> <span m="2689600">this</span> <span m="2690420">is</span> <span m="2690630">because</span> <span m="2692600">the</span> <span m="2692860">probability</span> <span m="2693860">that</span> <span m="2694020">you</span> <span m="2694160">kill</span> <span m="2694470">your</span> <span m="2694630">wife,</span> <span m="2698120">that's</span> <span m="2698400">K,</span> <span m="2700030">given</span> <span m="2700390">that</span> <span m="2700440">you</span> <span m="2700750">batter</span> <span m="2701210">your</span> <span m="2701360">wife,</span> <span m="2702350">that's</span> <span m="2702640">B,</span> <span m="2703830">is</span> <span m="2704060">1</span> <span m="2704350">in</span> <span m="2704480">2,000.</span> <span m="2707980">I</span> <span m="2708090">would</span> <span m="2708195">have</span> <span m="2708300">guessed</span> <span m="2708520">it</span> <span m="2708610">was</span> <span m="2708750">higher,</span> <span m="2709080">but</span> <span m="2710170">the</span> <span m="2710410">evidence</span> <span m="2710800">did</span> <span m="2710890">show</span> <span m="2711180">that.</span> <span m="2711680">And</span> <span m="2712180">so</span> <span m="2712400">they</span> <span m="2712520">said,</span> <span m="2712870">look,</span> <span m="2713070">there's</span> <span m="2713140">only</span> <span m="2713440">a</span> <span m="2713510">1</span> <span m="2713810">in</span> <span m="2713950">2,000</span> <span m="2714680">chance</span> <span m="2715250">that</span> <span m="2715350">this</span> <span m="2715590">evidence</span> <span m="2716080">of</span> <span m="2716410">wife</span> <span m="2716750">beating</span> <span m="2717580">is</span> <span m="2717750">relevant</span> <span m="2719640">and</span> <span m="2719920">therefore,</span> <span m="2721070">it</span> <span m="2721200">should</span> <span m="2721350">not</span> <span m="2721560">be</span> <span m="2721660">admitted</span> <span m="2722080">because</span> <span m="2723030">there's</span> <span m="2723190">a</span> <span m="2723260">pretty</span> <span m="2723460">decent</span> <span m="2723870">chance</span> <span m="2724180">if</span> <span m="2724250">the</span> <span m="2724340">jury</span> <span m="2724660">hears</span> <span m="2724940">this,</span> <span m="2725170">they're</span> <span m="2725280">going</span> <span m="2725390">to</span> <span m="2725450">convict</span> <span m="2725830">him.</span> </p>
<p><span m="2727340">That's</span> <span m="2727435">a</span> <span m="2727530">pretty</span> <span m="2727760">good</span> <span m="2727900">argument.</span> <span m="2729030">And</span> <span m="2729100">usually</span> <span m="2729380">that</span> <span m="2729540">kind</span> <span m="2729690">of</span> <span m="2729770">thing,</span> <span m="2729940">you</span> <span m="2730060">exclude</span> <span m="2730460">it.</span> <span m="2730660">Yeah.</span> </p>
<p><span m="2731090">AUDIENCE: Where</span> <span m="2731545">did</span> <span m="2732000">that</span> <span m="2732113">number</span> <span m="2732227">come</span> <span m="2732341">from?</span> </p>
<p><span m="2732455">TOM LEIGHTON:</span> <span m="2732910">They</span> <span m="2733140">got</span> <span m="2733400">some</span> <span m="2733630">study</span> <span m="2734110">and</span> <span m="2734250">some</span> <span m="2734430">experts</span> <span m="2734880">to</span> <span m="2734970">come</span> <span m="2735170">in</span> <span m="2735270">and</span> <span m="2735400">say</span> <span m="2735690">that</span> <span m="2737770">for</span> <span m="2737940">every</span> <span m="2738100">2,000</span> <span m="2738680">wife</span> <span m="2738940">beaters,</span> <span m="2739210">only</span> <span m="2739380">one</span> <span m="2739610">of</span> <span m="2739690">them</span> <span m="2739770">actually</span> <span m="2740070">kills</span> <span m="2740310">his</span> <span m="2740420">wife.</span> <span m="2744340">Now,</span> <span m="2745590">what</span> <span m="2745740">do</span> <span m="2746000">you</span> <span m="2746290">suppose</span> <span m="2746335">the</span> <span m="2746380">prosecution</span> <span m="2747190">argued</span> <span m="2747500">back?</span> <span m="2748290">They</span> <span m="2748410">actually</span> <span m="2748760">argued</span> <span m="2749050">back</span> <span m="2749270">very</span> <span m="2749470">effectively,</span> <span m="2749970">because</span> <span m="2750080">that's</span> <span m="2750750">a</span> <span m="2750860">tough</span> <span m="2751120">argument</span> <span m="2751470">to</span> <span m="2751530">get</span> <span m="2751690">by.</span> <span m="2753080">Yeah.</span> </p>
<p><span m="2753530">AUDIENCE: What's</span> <span m="2753998">the</span> <span m="2754466">probability</span> <span m="2754700">that</span> <span m="2754934">you</span> <span m="2755402">kill</span> <span m="2755870">your</span> <span m="2755987">wife</span> <span m="2756105">in</span> <span m="2756222">the</span> <span m="2756340">first</span> <span m="2756680">place,</span> <span m="2757380">that</span> <span m="2757700">could</span> <span m="2758077">be</span> <span m="2758831">100</span> <span m="2759210">times</span> <span m="2759560">larger</span> <span m="2759875">than</span> <span m="2760190">usual.</span> </p>
<p><span m="2760545">TOM LEIGHTON:</span> <span m="2760900">Well,</span> <span m="2761040">that's</span> <span m="2761270">a</span> <span m="2761300">good</span> <span m="2761530">point.</span> <span m="2762900">So</span> <span m="2763130">maybe</span> <span m="2763530">the</span> <span m="2763650">probability</span> <span m="2764240">of</span> <span m="2764340">killing</span> <span m="2764670">your</span> <span m="2764790">wife</span> <span m="2765210">not</span> <span m="2765490">knowing</span> <span m="2765840">B,</span> <span m="2767688">I</span> <span m="2768150">hope</span> <span m="2768370">is</span> <span m="2768510">pretty</span> <span m="2768720">small,</span> <span m="2770870">probably</span> <span m="2771370">that's</span> <span m="2771680">very</span> <span m="2772100">small,</span> <span m="2772470">but</span> <span m="2772760">I</span> <span m="2773060">don't</span> <span m="2773210">know.</span> <span m="2775810">But</span> <span m="2775950">in</span> <span m="2776010">any</span> <span m="2776210">case,</span> <span m="2776530">this</span> <span m="2776700">thing</span> <span m="2776880">you're</span> <span m="2776990">going</span> <span m="2777280">from,</span> <span m="2777450">say</span> <span m="2777630">it's</span> <span m="2777770">1</span> <span m="2777890">in</span> <span m="2778010">1</span> <span m="2778130">million</span> <span m="2778760">to</span> <span m="2779200">1</span> <span m="2779370">in</span> <span m="2779470">2,000,</span> <span m="2779855">1</span> <span m="2780240">in</span> <span m="2780340">2,000</span> <span m="2780650">is</span> <span m="2780960">still</span> <span m="2781220">too</span> <span m="2781390">small</span> <span m="2782420">to</span> <span m="2782500">be</span> <span m="2782600">used</span> <span m="2782805">as</span> <span m="2783010">evidence</span> <span m="2783560">that</span> <span m="2783770">OJ</span> <span m="2784200">did</span> <span m="2784642">it.</span> </p>
<p><span m="2785084">AUDIENCE:</span> <span m="2785526">Frequency</span> <span m="2785968">he</span> <span m="2786410">did</span> <span m="2786852">it.</span> </p>
<p><span m="2787294">TOM LEIGHTON:</span> <span m="2787740">Frequency,</span> <span m="2788055">they</span> <span m="2788370">didn't</span> <span m="2788680">get</span> <span m="2788800">into</span> <span m="2789010">that</span> <span m="2789220">because</span> <span m="2789450">I</span> <span m="2789573">guess</span> <span m="2789696">he'd</span> <span m="2789820">done</span> <span m="2790100">it</span> <span m="2790130">a</span> <span m="2790160">bunch,</span> <span m="2790570">but</span> <span m="2791210">that's</span> <span m="2791430">a</span> <span m="2791460">good</span> <span m="2791620">point.</span> <span m="2791840">It</span> <span m="2791910">could</span> <span m="2792090">be</span> <span m="2792290">there's</span> <span m="2792490">multiple</span> <span m="2792960">beatings</span> <span m="2793390">is</span> <span m="2793560">higher.</span> <span m="2794720">Maybe</span> <span m="2794940">that's</span> <span m="2795120">1</span> <span m="2795310">in</span> <span m="2795400">200</span> <span m="2795900">then.</span> <span m="2797500">In</span> <span m="2797590">fact,</span> <span m="2797770">that</span> <span m="2797930">may</span> <span m="2798045">be</span> <span m="2798160">the</span> <span m="2798280">case</span> <span m="2798570">because</span> <span m="2798890">I</span> <span m="2798960">think</span> <span m="2799160">there's</span> <span m="2799440">probably</span> <span m="2799820">they</span> <span m="2799930">say</span> <span m="2800130">because</span> <span m="2800330">if</span> <span m="2800460">you</span> <span m="2800540">do</span> <span m="2800700">it</span> <span m="2800770">once,</span> <span m="2800980">you</span> <span m="2801100">do</span> <span m="2801240">it</span> <span m="2801300">multiple</span> <span m="2801590">times.</span> <span m="2802870">So</span> <span m="2802980">there's</span> <span m="2803100">not</span> <span m="2803340">much</span> <span m="2803540">more</span> <span m="2803740">to</span> <span m="2803810">be</span> <span m="2803880">gaining</span> <span m="2804150">there.</span> </p>
<p><span m="2804940">There's</span> <span m="2805080">a</span> <span m="2805170">critical</span> <span m="2805890">piece</span> <span m="2806150">of</span> <span m="2806200">information</span> <span m="2806760">we've</span> <span m="2806950">left</span> <span m="2807320">out</span> <span m="2807480">of</span> <span m="2807580">our</span> <span m="2807700">conditional</span> <span m="2808140">probabilities</span> <span m="2808790">here.</span> <span m="2810276">In</span> <span m="2810690">fact,</span> <span m="2810850">the</span> <span m="2811010">most</span> <span m="2811160">glaring</span> <span m="2811580">piece</span> <span m="2812150">of</span> <span m="2812240">all</span> <span m="2812940">of</span> <span m="2813110">evidence.</span> <span m="2814500">What's</span> <span m="2814600">missing</span> <span m="2814980">here?</span> <span m="2815130">What</span> <span m="2815310">haven't</span> <span m="2815560">we</span> <span m="2815700">factored</span> <span m="2815990">in?</span> <span m="2816465">Yeah.</span> </p>
<p><span m="2816940">AUDIENCE:</span> <span m="2817415">The</span> <span m="2817890">probability</span> <span m="2818127">of</span> <span m="2818365">B.</span> </p>
<p><span m="2818840">TOM LEIGHTON: The</span> <span m="2819380">probability</span> <span m="2820180">of</span> <span m="2820520">B,</span> <span m="2820930">that's</span> <span m="2821150">the</span> <span m="2821250">battering.</span> <span m="2823490">Battering,</span> <span m="2824120">I</span> <span m="2824230">don't</span> <span m="2824360">know</span> <span m="2824460">what</span> <span m="2824580">it</span> <span m="2824680">is,</span> <span m="2824970">probably</span> <span m="2825200">a</span> <span m="2825280">large</span> <span m="2825660">number.</span> <span m="2827550">Defense</span> <span m="2827890">would</span> <span m="2828000">argue</span> <span m="2828310">it's</span> <span m="2828580">large,</span> <span m="2829000">I</span> <span m="2829050">guess,</span> <span m="2829450">but</span> <span m="2829860">it</span> <span m="2830100">shouldn't</span> <span m="2830260">matter</span> <span m="2831560">that</span> <span m="2831810">much.</span> </p>
<p><span m="2833664">AUDIENCE:</span> <span m="2834142">The</span> <span m="2834620">probability</span> <span m="2835098">that</span> <span m="2835576">he</span> <span m="2835815">actually</span> <span m="2836054">beat</span> <span m="2836532">her,</span> <span m="2837010">given</span> <span m="2837488">that</span> <span m="2837966">she</span> <span m="2838444">threatened</span> <span m="2838922">him?</span> </p>
<p><span m="2839400">TOM LEIGHTON:</span> <span m="2839890">Well,</span> <span m="2839990">there's</span> <span m="2840040">that,</span> <span m="2840130">but</span> <span m="2840760">they</span> <span m="2840850">have</span> <span m="2840960">police--</span> <span m="2841660">well,</span> <span m="2841850">that's</span> <span m="2842060">true.</span> <span m="2842160">They</span> <span m="2842280">didn't</span> <span m="2842470">see</span> <span m="2842710">him</span> <span m="2842820">doing</span> <span m="2843190">it,</span> <span m="2843300">but</span> <span m="2843380">let's</span> <span m="2843640">say</span> <span m="2843740">that</span> <span m="2843840">they</span> <span m="2844150">had</span> <span m="2844370">good</span> <span m="2844560">evidence</span> <span m="2844930">that</span> <span m="2845040">he</span> <span m="2845200">did</span> <span m="2845400">it</span> <span m="2845820">and</span> <span m="2847020">defense</span> <span m="2847470">wasn't</span> <span m="2847770">arguing</span> <span m="2847910">that</span> <span m="2848050">he</span> <span m="2848210">didn't</span> <span m="2848410">really</span> <span m="2848660">beat</span> <span m="2849000">her.</span> <span m="2850750">The</span> <span m="2850900">key</span> <span m="2851195">thing</span> <span m="2851490">we're</span> <span m="2851600">missing</span> <span m="2851950">here</span> <span m="2853120">is</span> <span m="2853410">Nicole</span> <span m="2853690">wound</span> <span m="2853960">up</span> <span m="2854110">dead.</span> <span m="2855770">She</span> <span m="2856000">was</span> <span m="2856160">dead.</span> <span m="2857780">And</span> <span m="2857930">there's</span> <span m="2858080">another</span> <span m="2858380">stat</span> <span m="2858760">here</span> <span m="2859010">that</span> <span m="2859260">the</span> <span m="2859470">prosecution</span> <span m="2860310">argued.</span> </p>
<p><span m="2872040">So</span> <span m="2872200">they</span> <span m="2872390">argued</span> <span m="2872680">this</span> <span m="2872900">fact.</span> <span m="2873730">The</span> <span m="2874230">probability</span> <span m="2875070">the</span> <span m="2875180">husband</span> <span m="2875610">kills</span> <span m="2875900">his</span> <span m="2876020">wife,</span> <span m="2876570">given</span> <span m="2876850">that</span> <span m="2877370">he</span> <span m="2877760">batters</span> <span m="2878230">her</span> <span m="2878610">and</span> <span m="2879390">she</span> <span m="2879590">wound</span> <span m="2879890">up</span> <span m="2880050">dead,</span> <span m="2880980">that</span> <span m="2881210">somebody</span> <span m="2881600">murder</span> <span m="2881965">her</span> <span m="2883040">is</span> <span m="2883240">bigger</span> <span m="2883550">than</span> <span m="2883700">1/2.</span> <span m="2885100">So</span> <span m="2885270">here</span> <span m="2885540">M</span> <span m="2885790">is</span> <span m="2885900">somebody</span> <span m="2886340">murdered</span> <span m="2886670">the</span> <span m="2886770">wife.</span> <span m="2888950">Here,</span> <span m="2889610">the</span> <span m="2889740">husband</span> <span m="2890090">beats</span> <span m="2890560">her.</span> <span m="2891030">Now,</span> <span m="2891530">the</span> <span m="2892250">conditional</span> <span m="2892570">probability</span> <span m="2892840">that</span> <span m="2892975">he</span> <span m="2893110">killed</span> <span m="2893345">her</span> <span m="2893580">is</span> <span m="2893710">bigger</span> <span m="2893940">than</span> <span m="2894070">1/2</span> <span m="2894710">and</span> <span m="2894880">that's</span> <span m="2895080">a</span> <span m="2895140">whopper.</span> <span m="2896540">Now,</span> <span m="2896790">it's</span> <span m="2896950">very</span> <span m="2897220">relevant.</span> </p>
<p><span m="2898760">The</span> <span m="2898830">probability</span> <span m="2899065">he</span> <span m="2899300">killed</span> <span m="2899500">her</span> <span m="2899700">just</span> <span m="2899920">given</span> <span m="2900120">that</span> <span m="2900190">he</span> <span m="2900455">beat</span> <span m="2900543">her</span> <span m="2900631">is</span> <span m="2900720">only</span> <span m="2900890">1</span> <span m="2901100">in</span> <span m="2901210">2,000,</span> <span m="2902290">but</span> <span m="2902470">if</span> <span m="2902550">you</span> <span m="2902650">add</span> <span m="2902960">the</span> <span m="2903040">fact,</span> <span m="2903590">which</span> <span m="2903810">is</span> <span m="2904120">very</span> <span m="2904350">relevant</span> <span m="2904680">in</span> <span m="2904740">this</span> <span m="2904920">case,</span> <span m="2905230">that</span> <span m="2905290">the</span> <span m="2905380">wife</span> <span m="2905630">was</span> <span m="2906000">murdered,</span> <span m="2907730">this</span> <span m="2907950">is</span> <span m="2908060">now</span> <span m="2908870">very</span> <span m="2909130">compelling.</span> <span m="2910290">Now,</span> <span m="2910330">in</span> <span m="2910430">fact,</span> <span m="2910700">they</span> <span m="2910780">should</span> <span m="2910970">have</span> <span m="2911160">really</span> <span m="2911490">compare</span> <span m="2911960">this</span> <span m="2913160">to</span> <span m="2914650">probability</span> <span m="2916250">he</span> <span m="2916350">kills</span> <span m="2916850">her</span> <span m="2917520">given</span> <span m="2917790">that</span> <span m="2918030">she's</span> <span m="2918300">dead.</span> <span m="2919840">And</span> <span m="2920020">so</span> <span m="2920120">that</span> <span m="2920340">would</span> <span m="2920720">determine</span> <span m="2921110">now</span> <span m="2921220">the</span> <span m="2921330">relevance</span> <span m="2921750">of</span> <span m="2923070">the</span> <span m="2923170">battering,</span> <span m="2923430">the</span> <span m="2924370">wife</span> <span m="2924630">beating.</span> <span m="2925490">That's</span> <span m="2925690">what</span> <span m="2925810">they</span> <span m="2925890">should</span> <span m="2926220">have</span> <span m="2926360">done,</span> <span m="2926610">but</span> <span m="2926750">they</span> <span m="2927145">didn't.</span> <span m="2927342">They</span> <span m="2927540">got</span> <span m="2927780">this</span> <span m="2927990">far</span> <span m="2928480">and</span> <span m="2928690">they</span> <span m="2928810">had</span> <span m="2929000">that</span> <span m="2929220">and</span> <span m="2929270">the</span> <span m="2929320">judge</span> <span m="2929590">said,</span> <span m="2929845">I'm</span> <span m="2930100">letting</span> <span m="2930270">it</span> <span m="2930610">in.</span> <span m="2931610">So</span> <span m="2931770">it</span> <span m="2931940">came</span> <span m="2932180">in</span> <span m="2932390">at</span> <span m="2932490">that</span> <span m="2932670">point.</span> </p>
<p><span m="2933450">But</span> <span m="2933560">this</span> <span m="2933700">would</span> <span m="2933780">be</span> <span m="2933860">the</span> <span m="2933970">right</span> <span m="2934250">comparison,</span> <span m="2934780">I</span> <span m="2934850">think.</span> <span m="2935600">Because</span> <span m="2935870">you</span> <span m="2936000">look</span> <span m="2936190">at</span> <span m="2936250">the</span> <span m="2936310">probability</span> <span m="2936880">that</span> <span m="2937010">you</span> <span m="2937410">killed</span> <span m="2937860">her</span> <span m="2938980">given</span> <span m="2939240">that</span> <span m="2939330">she's</span> <span m="2939520">dead,</span> <span m="2940650">but</span> <span m="2940780">now</span> <span m="2940940">the</span> <span m="2941040">additional</span> <span m="2941360">information,</span> <span m="2941790">the</span> <span m="2941860">wife</span> <span m="2942120">battering,</span> <span m="2942420">how</span> <span m="2942600">does</span> <span m="2942700">that</span> <span m="2942850">change</span> <span m="2943200">the</span> <span m="2943280">probability?</span> <span m="2944310">And</span> <span m="2944430">it</span> <span m="2944500">probably</span> <span m="2944750">changes</span> <span m="2945100">it</span> <span m="2945310">materially.</span> <span m="2948480">So</span> <span m="2948590">it's</span> <span m="2948690">all</span> <span m="2948800">a</span> <span m="2948910">little</span> <span m="2949110">gory,</span> <span m="2949970">but</span> <span m="2950180">it's</span> <span m="2950450">interesting</span> <span m="2951150">to</span> <span m="2951250">see</span> <span m="2951420">how</span> <span m="2951530">mathematics</span> <span m="2952990">played</span> <span m="2953210">out</span> <span m="2953780">in</span> <span m="2953850">this</span> <span m="2953990">kind</span> <span m="2954130">of</span> <span m="2954210">environment.</span> <span m="2954790">Yeah.</span> </p>
<p><span m="2955190">AUDIENCE:</span> <span m="2955643">Are</span> <span m="2955733">we</span> <span m="2955824">supposed</span> <span m="2955914">to</span> <span m="2956005">assume</span> <span m="2956096">that</span> <span m="2956549">he</span> <span m="2957002">did</span> <span m="2957455">kill</span> <span m="2957908">his</span> <span m="2958361">wife?</span> </p>
<p><span m="2958814">TOM LEIGHTON:</span> <span m="2959270">Yes,</span> <span m="2959700">and</span> <span m="2959770">they</span> <span m="2959840">assumed</span> <span m="2960300">that,</span> <span m="2960700">but</span> <span m="2961100">when</span> <span m="2961420">you</span> <span m="2961520">decide</span> <span m="2962130">whether</span> <span m="2962330">or</span> <span m="2962360">not</span> <span m="2962540">to</span> <span m="2962740">admit</span> <span m="2962920">evidence,</span> <span m="2964170">if</span> <span m="2964600">it's</span> <span m="2965030">prejudicial,</span> <span m="2966570">you've</span> <span m="2966580">got</span> <span m="2966690">to</span> <span m="2966740">have</span> <span m="2966840">a</span> <span m="2966950">really</span> <span m="2967310">good</span> <span m="2967480">grounds</span> <span m="2967850">to</span> <span m="2967930">get</span> <span m="2968100">it</span> <span m="2968190">in.</span> <span m="2968410">Like</span> <span m="2968620">if</span> <span m="2968780">the</span> <span m="2968940">evidence</span> <span m="2969340">is</span> <span m="2969440">going</span> <span m="2969550">to</span> <span m="2969590">make</span> <span m="2969830">the</span> <span m="2969900">jury</span> <span m="2970230">think</span> <span m="2970450">he</span> <span m="2970570">did</span> <span m="2970790">it,</span> <span m="2971710">then</span> <span m="2971850">you</span> <span m="2971970">really</span> <span m="2972320">got</span> <span m="2972395">to</span> <span m="2972470">argue</span> <span m="2972820">the</span> <span m="2972970">evidence</span> <span m="2973330">is</span> <span m="2973450">relevant</span> <span m="2974340">somehow.</span> <span m="2975480">There's</span> <span m="2975830">material</span> <span m="2976380">information</span> <span m="2977560">and</span> <span m="2977720">that's</span> <span m="2977890">what</span> <span m="2977970">the</span> <span m="2978050">fight</span> <span m="2978310">was</span> <span m="2978440">about.</span> <span m="2978690">A</span> <span m="2978750">1</span> <span m="2978980">in</span> <span m="2979070">2,000</span> <span m="2979640">relevance</span> <span m="2980120">isn't</span> <span m="2980410">going</span> <span m="2980510">to</span> <span m="2980620">cut</span> <span m="2980915">it.</span> <span m="2981740">1</span> <span m="2982900">in</span> <span m="2983030">2,</span> <span m="2983930">that's</span> <span m="2984180">probably</span> <span m="2984430">pretty</span> <span m="2984620">relevant.</span> <span m="2985720">And</span> <span m="2985840">that</span> <span m="2986050">will</span> <span m="2986230">be</span> <span m="2986400">the</span> <span m="2986480">grounds</span> <span m="2986870">on</span> <span m="2986910">which</span> <span m="2987110">the</span> <span m="2987210">judge</span> <span m="2987500">makes</span> <span m="2987760">his</span> <span m="2987870">decision.</span> <span m="2989460">But</span> <span m="2989590">yeah,</span> <span m="2989760">you</span> <span m="2989900">assume</span> <span m="2990180">he</span> <span m="2990310">didn't</span> <span m="2990440">do</span> <span m="2990690">it.</span> </p>
<p><span m="2996256">All</span> <span m="2996640">right.</span> <span m="2996860">Back</span> <span m="2997020">to</span> <span m="2997180">independence.</span> <span m="2997830">So</span> <span m="2998440">the</span> <span m="2998560">last</span> <span m="2998870">example</span> <span m="2999310">today</span> <span m="3000250">is</span> <span m="3000700">derived</span> <span m="3001100">from</span> <span m="3001230">a</span> <span m="3001280">famous</span> <span m="3001660">paradox</span> <span m="3003410">and</span> <span m="3003530">has</span> <span m="3003690">several</span> <span m="3004180">actually</span> <span m="3004420">important</span> <span m="3004860">applications</span> <span m="3005470">in</span> <span m="3005520">computer</span> <span m="3005880">science.</span> <span m="3006880">And</span> <span m="3007060">this</span> <span m="3007260">problem</span> <span m="3007530">is</span> <span m="3007650">known</span> <span m="3007840">as</span> <span m="3007950">the</span> <span m="3008050">birthday</span> <span m="3008850">problem</span> <span m="3009370">or</span> <span m="3009450">the</span> <span m="3009560">birthday</span> <span m="3009960">paradox.</span> <span m="3013170">It's</span> <span m="3013540">a</span> <span m="3013640">paradox</span> <span m="3014180">because</span> <span m="3014590">it</span> <span m="3014750">sort</span> <span m="3014790">of</span> <span m="3014830">has</span> <span m="3015030">a</span> <span m="3015090">surprising</span> <span m="3015700">answer.</span> <span m="3020220">Probably</span> <span m="3020520">a</span> <span m="3020590">lot</span> <span m="3020870">of</span> <span m="3020940">you</span> <span m="3021130">have</span> <span m="3021230">seen</span> <span m="3021440">this</span> <span m="3021590">before</span> <span m="3021980">in</span> <span m="3022050">some</span> <span m="3022350">form</span> <span m="3022592">or</span> <span m="3022834">another.</span> </p>
<p><span m="3037860">In</span> <span m="3037970">the</span> <span m="3038050">birthday</span> <span m="3038390">problem,</span> <span m="3038900">there</span> <span m="3039110">are</span> <span m="3039530">N</span> <span m="3039990">birthdays</span> <span m="3045180">and</span> <span m="3045880">typically</span> <span m="3046310">we're</span> <span m="3046420">going</span> <span m="3046600">to</span> <span m="3046640">look</span> <span m="3046680">at</span> <span m="3046720">the</span> <span m="3046760">case</span> <span m="3047090">where</span> <span m="3048780">N</span> <span m="3049040">is</span> <span m="3049210">365,</span> <span m="3050470">the</span> <span m="3050970">days</span> <span m="3051280">of</span> <span m="3051380">the</span> <span m="3052220">year,</span> <span m="3053290">and</span> <span m="3053480">there</span> <span m="3053620">is</span> <span m="3053760">M</span> <span m="3054020">people.</span> <span m="3059330">And</span> <span m="3059540">for</span> <span m="3059650">example,</span> <span m="3061250">know</span> <span m="3061330">maybe</span> <span m="3061520">there's</span> <span m="3061650">100</span> <span m="3061990">people</span> <span m="3062280">here.</span> <span m="3067070">And</span> <span m="3067260">what</span> <span m="3067350">we</span> <span m="3067460">want</span> <span m="3067680">to</span> <span m="3067730">know</span> <span m="3069640">is,</span> <span m="3069990">what</span> <span m="3070140">is</span> <span m="3070240">the</span> <span m="3070340">probability</span> <span m="3075480">that</span> <span m="3075810">two</span> <span m="3076000">or</span> <span m="3076040">more</span> <span m="3076270">people</span> <span m="3080960">have</span> <span m="3081270">the</span> <span m="3081370">same</span> <span m="3081490">birthday.</span> </p>
<p><span m="3092760">For</span> <span m="3092880">example,</span> <span m="3093380">how</span> <span m="3093430">many</span> <span m="3093640">people</span> <span m="3093980">think</span> <span m="3094260">there's</span> <span m="3094450">at</span> <span m="3094570">least</span> <span m="3094870">a</span> <span m="3094930">50%</span> <span m="3095690">chance</span> <span m="3096200">that</span> <span m="3096610">a</span> <span m="3096710">pair</span> <span m="3097100">of</span> <span m="3097180">you</span> <span m="3097450">in</span> <span m="3097530">the</span> <span m="3097680">audience</span> <span m="3098080">here</span> <span m="3098360">have</span> <span m="3098650">the</span> <span m="3098720">same</span> <span m="3098990">birthday?</span> <span m="3101460">That's</span> <span m="3102090">good.</span> <span m="3102870">How</span> <span m="3102980">many</span> <span m="3103180">people</span> <span m="3103510">think</span> <span m="3103950">there's</span> <span m="3104180">a</span> <span m="3104240">better</span> <span m="3104980">than</span> <span m="3105280">90%</span> <span m="3106110">chance?</span> <span m="3108030">A</span> <span m="3108300">few</span> <span m="3108345">of</span> <span m="3108390">you.</span> <span m="3109382">All</span> <span m="3109760">right.</span> <span m="3109930">How</span> <span m="3110070">many</span> <span m="3110260">people</span> <span m="3110550">think</span> <span m="3110720">there's</span> <span m="3110860">a</span> <span m="3110920">better</span> <span m="3111270">than</span> <span m="3111420">a</span> <span m="3111620">99%</span> <span m="3112810">chance</span> <span m="3113410">that</span> <span m="3113780">there's</span> <span m="3113930">a</span> <span m="3113990">pair</span> <span m="3114300">of</span> <span m="3114350">matching</span> <span m="3114680">birthdays?</span> <span m="3115700">A</span> <span m="3115780">couple</span> <span m="3116110">left.</span> </p>
<p><span m="3116950">How</span> <span m="3117120">many</span> <span m="3117330">think</span> <span m="3117520">it's</span> <span m="3117680">better</span> <span m="3117890">than</span> <span m="3118030">a</span> <span m="3118100">99.9%</span> <span m="3120060">chance?</span> <span m="3121060">We've</span> <span m="3121310">got</span> <span m="3121560">one,</span> <span m="3122050">two.</span> <span m="3122440">You</span> <span m="3122570">guys</span> <span m="3122770">are</span> <span m="3122800">going</span> <span m="3122830">to</span> <span m="3122860">be</span> <span m="3122950">stubborn.</span> <span m="3123390">Another</span> <span m="3123630">one.</span> <span m="3123990">All</span> <span m="3124330">right.</span> <span m="3124500">How</span> <span m="3124630">many</span> <span m="3124780">people</span> <span m="3125060">think</span> <span m="3125360">it's</span> <span m="3125600">more</span> <span m="3125830">than</span> <span m="3126010">99.999%</span> <span m="3129580">chance?</span> <span m="3131770">Actually</span> <span m="3132120">it's</span> <span m="3132430">six</span> <span m="3132885">9's.</span> <span m="3133340">It's</span> <span m="3133900">incredible.</span> <span m="3135440">It</span> <span m="3135560">is</span> <span m="3135700">a</span> <span m="3135750">virtual</span> <span m="3136190">certainty.</span> </p>
<p><span m="3138120">So</span> <span m="3138760">let's</span> <span m="3139050">see.</span> <span m="3139200">In</span> <span m="3139290">fact,</span> <span m="3139880">the</span> <span m="3139970">chance</span> <span m="3140390">that</span> <span m="3140520">you're</span> <span m="3140710">all</span> <span m="3140920">different</span> <span m="3142440">is</span> <span m="3142640">about</span> <span m="3142930">1</span> <span m="3143170">in</span> <span m="3143280">3</span> <span m="3143610">million</span> <span m="3144970">chance</span> <span m="3145380">that</span> <span m="3145495">you're</span> <span m="3145610">all</span> <span m="3145820">different.</span> <span m="3147290">And</span> <span m="3147420">we're</span> <span m="3147550">going</span> <span m="3147650">to</span> <span m="3148020">see</span> <span m="3148270">why</span> <span m="3148490">that's</span> <span m="3148990">true</span> <span m="3149300">here.</span> <span m="3150810">But</span> <span m="3151070">to</span> <span m="3151170">do</span> <span m="3151420">that,</span> <span m="3151890">we're</span> <span m="3152030">going</span> <span m="3152120">to</span> <span m="3152190">need</span> <span m="3152460">to</span> <span m="3152540">make</span> <span m="3153410">two</span> <span m="3153910">important</span> <span m="3154440">assumptions.</span> <span m="3156720">Any</span> <span m="3156940">ideas</span> <span m="3157440">about</span> <span m="3157870">what</span> <span m="3158020">assumptions</span> <span m="3158490">you're</span> <span m="3158550">going</span> <span m="3158650">to</span> <span m="3158690">need?</span> <span m="3159980">Yeah.</span> </p>
<p><span m="3160350">AUDIENCE: Birthdays</span> <span m="3160805">are</span> <span m="3161260">uniformly</span> <span m="3161715">distributed.</span> </p>
<p><span m="3162170">TOM LEIGHTON: Birthdays</span> <span m="3162630">are</span> <span m="3162670">uniformly</span> <span m="3163260">distributed.</span> <span m="3163585">Any</span> <span m="3163910">other</span> <span m="3164346">ideas?</span> <span m="3164782">Yes.</span> </p>
<p><span m="3165218">AUDIENCE: He</span> <span m="3165654">stole</span> <span m="3166090">my</span> <span m="3166310">answer.</span> </p>
<p><span m="3166530">TOM LEIGHTON: Oh,</span> <span m="3166810">he</span> <span m="3166896">stole</span> <span m="3166983">yours.</span> <span m="3167870">What</span> <span m="3168195">else</span> <span m="3168260">are</span> <span m="3168325">you</span> <span m="3168390">going</span> <span m="3168455">to</span> <span m="3168520">need</span> <span m="3168650">to</span> <span m="3168780">assume?</span> <span m="3170724">Yeah.</span> </p>
<p><span m="3171700">AUDIENCE:</span> <span m="3172190">All</span> <span m="3172360">birthdays</span> <span m="3172960">are</span> <span m="3173780">independent</span> <span m="3174190">of</span> <span m="3174280">each</span> <span m="3174410">other.</span> </p>
<p><span m="3174720">TOM LEIGHTON: Yeah.</span> <span m="3175110">Mutually</span> <span m="3175650">independent.</span> <span m="3176410">We're</span> <span m="3176530">going</span> <span m="3176630">to</span> <span m="3176680">need</span> <span m="3176820">that</span> <span m="3177070">as</span> <span m="3177190">well.</span> <span m="3178380">Now,</span> <span m="3178880">in</span> <span m="3179000">actuality,</span> <span m="3179640">neither</span> <span m="3180580">is</span> <span m="3180760">true</span> <span m="3181680">in</span> <span m="3181830">reality.</span> <span m="3183020">It's</span> <span m="3183220">well</span> <span m="3183550">known</span> <span m="3183900">that</span> <span m="3183980">birthdays</span> <span m="3184610">tend</span> <span m="3184870">to</span> <span m="3184960">follow</span> <span m="3185520">seasonal</span> <span m="3186250">patterns</span> <span m="3187320">and</span> <span m="3187550">they're</span> <span m="3187700">related</span> <span m="3188260">to</span> <span m="3188360">major</span> <span m="3188780">events.</span> </p>
<p><span m="3190070">Now,</span> <span m="3190670">do</span> <span m="3190880">you</span> <span m="3191090">all</span> <span m="3191210">remember</span> <span m="3191610">the</span> <span m="3191730">big</span> <span m="3192240">blackout</span> <span m="3192750">that</span> <span m="3192820">hit</span> <span m="3192980">the</span> <span m="3193060">Northeast</span> <span m="3193590">several</span> <span m="3193940">years</span> <span m="3194160">ago?</span> <span m="3194830">Do</span> <span m="3195190">you</span> <span m="3195550">remember</span> <span m="3195790">that?</span> <span m="3196710">Well,</span> <span m="3196900">it</span> <span m="3197000">turns</span> <span m="3197310">out,</span> <span m="3197650">this</span> <span m="3197860">is</span> <span m="3198120">a</span> <span m="3198230">true</span> <span m="3198470">fact,</span> <span m="3198660">there</span> <span m="3198850">were</span> <span m="3198960">a</span> <span m="3199010">lot</span> <span m="3199190">of</span> <span m="3199280">babies</span> <span m="3199700">born</span> <span m="3200000">nine</span> <span m="3200230">months</span> <span m="3200480">later.</span> <span m="3201850">In</span> <span m="3202140">fact,</span> <span m="3202580">they</span> <span m="3202690">had</span> <span m="3202850">a</span> <span m="3202960">name.</span> <span m="3203020">They're</span> <span m="3203140">called</span> <span m="3203320">blackout</span> <span m="3203810">babies.</span> <span m="3204730">If</span> <span m="3204880">you</span> <span m="3205000">were</span> <span m="3205080">born</span> <span m="3205570">in</span> <span m="3206020">that</span> <span m="3206240">period</span> <span m="3206520">in</span> <span m="3206600">the</span> <span m="3206670">Northeast</span> <span m="3207700">and</span> <span m="3207810">there's</span> <span m="3207950">all</span> <span m="3208140">these</span> <span m="3208350">news</span> <span m="3208720">stories</span> <span m="3209190">about</span> <span m="3209550">the</span> <span m="3209640">life</span> <span m="3209930">of</span> <span m="3210030">the</span> <span m="3210100">blackout</span> <span m="3210560">babies.</span> </p>
<p><span m="3211770">And</span> <span m="3211920">the</span> <span m="3212000">same</span> <span m="3212240">thing</span> <span m="3212400">happens</span> <span m="3212810">after</span> <span m="3213060">cold</span> <span m="3213430">snaps</span> <span m="3213840">in</span> <span m="3213890">the</span> <span m="3213980">winter</span> <span m="3214370">and</span> <span m="3214760">you</span> <span m="3214840">get</span> <span m="3214960">a</span> <span m="3215000">blizzard</span> <span m="3215400">or</span> <span m="3215440">this</span> <span m="3215590">kind</span> <span m="3215740">of</span> <span m="3215820">a</span> <span m="3215880">thing.</span> <span m="3216860">Nine</span> <span m="3217100">months</span> <span m="3217340">later,</span> <span m="3218160">you</span> <span m="3218260">get</span> <span m="3218450">babies.</span> <span m="3219150">In</span> <span m="3219250">fact,</span> <span m="3220620">I</span> <span m="3220800">had</span> <span m="3220930">a</span> <span m="3220990">personal</span> <span m="3221370">experience</span> <span m="3221950">with</span> <span m="3222090">this.</span> <span m="3224040">Well,</span> <span m="3226740">my</span> <span m="3226880">son</span> <span m="3227280">was</span> <span m="3227470">born</span> <span m="3228330">on</span> <span m="3228460">October</span> <span m="3229230">18,</span> <span m="3229990">1996.</span> <span m="3231090">And</span> <span m="3231280">on</span> <span m="3231440">the</span> <span m="3231540">day</span> <span m="3231880">he</span> <span m="3232010">was</span> <span m="3232180">born,</span> <span m="3232470">we're</span> <span m="3232560">going</span> <span m="3232780">to</span> <span m="3232820">the</span> <span m="3232920">hospital</span> <span m="3234100">and</span> <span m="3234420">it</span> <span m="3234540">was</span> <span m="3234740">a</span> <span m="3234810">zoo.</span> </p>
<p><span m="3235720">The</span> <span m="3235820">maternity</span> <span m="3236260">ward</span> <span m="3236620">was</span> <span m="3236770">totally</span> <span m="3237260">full.</span> <span m="3238190">We</span> <span m="3238500">had</span> <span m="3238720">to</span> <span m="3238780">go</span> <span m="3239120">at</span> <span m="3239210">some</span> <span m="3239370">other</span> <span m="3239530">wing</span> <span m="3239720">of</span> <span m="3239780">the</span> <span m="3239840">hospital.</span> <span m="3241050">And</span> <span m="3242230">babies</span> <span m="3242630">were</span> <span m="3242740">popping</span> <span m="3243140">out</span> <span m="3243290">all</span> <span m="3243470">over</span> <span m="3243590">the</span> <span m="3243690">place.</span> <span m="3245300">And</span> <span m="3245480">I</span> <span m="3245550">asked,</span> <span m="3245810">what</span> <span m="3246140">is</span> <span m="3246330">going</span> <span m="3246620">on?</span> <span m="3247040">Why</span> <span m="3247570">don't</span> <span m="3247730">you</span> <span m="3247800">have</span> <span m="3247890">enough</span> <span m="3248070">room</span> <span m="3248920">for</span> <span m="3249120">all</span> <span m="3249330">the</span> <span m="3249390">mothers</span> <span m="3249750">here?</span> </p>
<p><span m="3250540">And</span> <span m="3250650">they</span> <span m="3250740">said,</span> <span m="3250950">oh,</span> <span m="3251120">it's</span> <span m="3251230">all</span> <span m="3251380">the</span> <span m="3251460">blizzard</span> <span m="3251830">babies.</span> <span m="3252520">And</span> <span m="3252790">I</span> <span m="3252930">go,</span> <span m="3253050">what?</span> <span m="3253790">And</span> <span m="3253930">they</span> <span m="3253990">go,</span> <span m="3254190">well,</span> <span m="3254390">remember</span> <span m="3254650">the</span> <span m="3254750">blizzard</span> <span m="3255100">of</span> <span m="3255170">'96?</span> <span m="3256100">It's</span> <span m="3256440">like,</span> <span m="3256630">oh</span> <span m="3257113">yeah.</span> <span m="3258562">I</span> <span m="3259045">remember.</span> <span m="3259528">Yeah.</span> <span m="3260980">It</span> <span m="3261455">was</span> <span m="3261930">nine</span> <span m="3262230">months</span> <span m="3262520">prior</span> <span m="3263080">is</span> <span m="3263220">the</span> <span m="3263310">big</span> <span m="3263500">blizzard</span> <span m="3263930">and</span> <span m="3264050">so</span> <span m="3264210">it's</span> <span m="3264330">all</span> <span m="3264560">the</span> <span m="3264650">blizzard</span> <span m="3265070">babies</span> <span m="3265370">coming.</span> </p>
<p><span m="3268000">So</span> <span m="3269260">they're</span> <span m="3269410">not</span> <span m="3269740">uniform.</span> <span m="3270310">They're</span> <span m="3270450">all</span> <span m="3270780">different</span> <span m="3271050">probabilities</span> <span m="3271690">here,</span> <span m="3271830">but</span> <span m="3271980">we're</span> <span m="3272170">going</span> <span m="3272260">to</span> <span m="3272350">assume</span> <span m="3273180">they're</span> <span m="3273340">equally</span> <span m="3273650">likely.</span> </p>
<p><span m="3276570">Now,</span> <span m="3278120">independence</span> <span m="3278860">is</span> <span m="3279020">also</span> <span m="3279510">not</span> <span m="3279790">true,</span> <span m="3280660">in</span> <span m="3280830">general.</span> <span m="3281920">What's</span> <span m="3282300">one</span> <span m="3282620">way</span> <span m="3283270">that</span> <span m="3283390">birthdays</span> <span m="3283820">might</span> <span m="3284210">not</span> <span m="3284510">be</span> <span m="3284770">independent?</span> <span m="3287210">What</span> <span m="3287380">is</span> <span m="3287520">it?</span> </p>
<p><span m="3287935">AUDIENCE: Twins.</span> </p>
<p><span m="3288350">TOM LEIGHTON: Twins.</span> <span m="3289540">So</span> <span m="3289655">if</span> <span m="3289770">they're</span> <span m="3289920">twins,</span> <span m="3290300">they</span> <span m="3290615">have</span> <span m="3290930">the</span> <span m="3291020">same</span> <span m="3291260">birthday.</span> <span m="3292530">Now,</span> <span m="3292640">there's</span> <span m="3292840">other</span> <span m="3293050">ways.</span> <span m="3293500">In</span> <span m="3293610">fact,</span> <span m="3294570">my</span> <span m="3295190">only</span> <span m="3295510">sibling,</span> <span m="3295900">my</span> <span m="3296030">brother,</span> <span m="3296990">has</span> <span m="3297260">the</span> <span m="3297370">same</span> <span m="3297770">birthday</span> <span m="3298150">I</span> <span m="3298300">do,</span> <span m="3299130">but</span> <span m="3299260">I'm</span> <span m="3299430">two</span> <span m="3299690">years</span> <span m="3300060">older,</span> <span m="3300880">so</span> <span m="3301030">we</span> <span m="3301160">weren't</span> <span m="3301600">twins.</span> <span m="3302680">Now,</span> <span m="3302790">you</span> <span m="3302910">say,</span> <span m="3303700">what</span> <span m="3303890">are</span> <span m="3303920">the</span> <span m="3304100">odds</span> <span m="3304370">of</span> <span m="3304430">that?</span> <span m="3305780">Well,</span> <span m="3306050">1</span> <span m="3306300">in</span> <span m="3306440">365,</span> <span m="3308010">you</span> <span m="3308140">think.</span> </p>
<p><span m="3310200">Well,</span> <span m="3310420">one</span> <span m="3310890">day</span> <span m="3311090">I'm</span> <span m="3311210">in</span> <span m="3311300">middle</span> <span m="3311550">school,</span> <span m="3311800">about</span> <span m="3311970">the</span> <span m="3312120">age</span> <span m="3312290">you</span> <span m="3312350">start</span> <span m="3312620">thinking</span> <span m="3312840">about</span> <span m="3313070">these</span> <span m="3313290">things,</span> <span m="3314040">and</span> <span m="3314200">you</span> <span m="3314250">get</span> <span m="3314390">the</span> <span m="3314500">idea</span> <span m="3314780">to</span> <span m="3314870">count</span> <span m="3315170">back</span> <span m="3315440">nine</span> <span m="3315720">months</span> <span m="3316060">from</span> <span m="3316460">your</span> <span m="3316660">birthday.</span> <span m="3317160">Probably</span> <span m="3317600">some</span> <span m="3317657">of</span> <span m="3317715">you</span> <span m="3317772">have</span> <span m="3317830">done</span> <span m="3318040">that.</span> <span m="3318940">And</span> <span m="3320130">I</span> <span m="3320180">did</span> <span m="3320420">that</span> <span m="3320610">and</span> <span m="3320870">that's</span> <span m="3321310">my</span> <span m="3321400">dad's</span> <span m="3321830">birthday.</span> <span m="3324490">I</span> <span m="3324560">was</span> <span m="3324750">like,</span> <span m="3325480">oh.</span> <span m="3328520">May</span> <span m="3329230">is</span> <span m="3329400">not</span> <span m="3329750">1</span> <span m="3329990">in</span> <span m="3330140">365.</span> <span m="3332100">It's</span> <span m="3332270">like,</span> <span m="3333310">Happy</span> <span m="3333640">Birthday.</span> <span m="3335810">I</span> <span m="3336140">don't</span> <span m="3336270">know.</span> </p>
<p><span m="3336860">Anyway,</span> <span m="3337170">I</span> <span m="3337310">almost</span> <span m="3337510">needed</span> <span m="3337750">to</span> <span m="3337800">go</span> <span m="3337960">into</span> <span m="3338090">therapy</span> <span m="3338600">after</span> <span m="3338850">that,</span> <span m="3339160">you</span> <span m="3339250">know.</span> <span m="3340800">So</span> <span m="3341710">now</span> <span m="3341880">you</span> <span m="3342160">all</span> <span m="3342193">got</span> <span m="3342226">to</span> <span m="3342260">count</span> <span m="3342560">back</span> <span m="3342800">nine</span> <span m="3343090">months</span> <span m="3343370">from</span> <span m="3343440">your</span> <span m="3343530">birthday.</span> <span m="3345810">Anybody</span> <span m="3346040">whose</span> <span m="3346270">birthday</span> <span m="3346690">is</span> <span m="3346745">on</span> <span m="3346800">September</span> <span m="3347310">30</span> <span m="3347820">or</span> <span m="3348080">October</span> <span m="3348310">1,</span> <span m="3349100">nine</span> <span m="3349370">months</span> <span m="3349610">back</span> <span m="3349870">is</span> <span m="3349990">New</span> <span m="3350090">Year's</span> <span m="3350340">Eve.</span> <span m="3351360">That's</span> <span m="3351540">dangerous.</span> <span m="3353100">So</span> <span m="3353620">in</span> <span m="3353730">reality,</span> <span m="3355060">birthdays</span> <span m="3355510">are</span> <span m="3355610">not</span> <span m="3356180">independent</span> <span m="3357100">and</span> <span m="3357320">they</span> <span m="3357440">are</span> <span m="3357530">not</span> <span m="3358230">randomly</span> <span m="3358820">distributed,</span> <span m="3359610">but</span> <span m="3359800">we're</span> <span m="3359950">going</span> <span m="3360050">to</span> <span m="3360110">assume</span> <span m="3360630">that</span> <span m="3360970">because</span> <span m="3362040">we're</span> <span m="3362280">going</span> <span m="3362370">to</span> <span m="3362410">use</span> <span m="3362760">this</span> <span m="3362890">same</span> <span m="3363360">analysis</span> <span m="3364190">for</span> <span m="3364340">computer</span> <span m="3364700">science</span> <span m="3365090">problems</span> <span m="3365510">where</span> <span m="3365650">things</span> <span m="3365930">are,</span> <span m="3366550">hopefully,</span> <span m="3367020">more</span> <span m="3367760">independent</span> <span m="3368410">and</span> <span m="3368560">random.</span> </p>
<p><span m="3369990">Now,</span> <span m="3370120">we're</span> <span m="3370260">going</span> <span m="3370370">to</span> <span m="3370430">do</span> <span m="3370590">an</span> <span m="3370660">experiment</span> <span m="3371330">to</span> <span m="3371430">see</span> <span m="3371660">how</span> <span m="3371870">many</span> <span m="3372190">people</span> <span m="3372590">it</span> <span m="3372710">takes</span> <span m="3373060">us</span> <span m="3373970">to</span> <span m="3374100">get</span> <span m="3374620">a</span> <span m="3374710">pair</span> <span m="3374930">of</span> <span m="3374970">matching</span> <span m="3375290">birthdays.</span> <span m="3375710">So</span> <span m="3375860">I'm</span> <span m="3375920">going</span> <span m="3376030">to</span> <span m="3376210">run</span> <span m="3376440">through</span> <span m="3376610">people</span> <span m="3377050">in</span> <span m="3377440">order</span> <span m="3377770">in</span> <span m="3377835">the</span> <span m="3377900">rows</span> <span m="3378220">here,</span> <span m="3379000">get</span> <span m="3379170">your</span> <span m="3379290">birthday</span> <span m="3379510">and</span> <span m="3379730">we're</span> <span m="3379900">going</span> <span m="3379990">to</span> <span m="3380030">record</span> <span m="3380440">and</span> <span m="3380500">we're</span> <span m="3380595">going</span> <span m="3380690">to</span> <span m="3380760">see</span> <span m="3380900">how</span> <span m="3381140">far</span> <span m="3381380">we</span> <span m="3381490">go</span> <span m="3382260">until</span> <span m="3382380">there's</span> <span m="3382680">a</span> <span m="3382740">match</span> <span m="3383320">in</span> <span m="3383440">that</span> <span m="3383640">group.</span> <span m="3384880">So</span> <span m="3384970">I</span> <span m="3385020">will</span> <span m="3385190">write</span> <span m="3385360">up</span> <span m="3385440">the</span> <span m="3385540">months</span> <span m="3385910">here.</span> <span m="3411230">And</span> <span m="3411480">we'll</span> <span m="3411590">start</span> <span m="3411900">with</span> <span m="3411990">my</span> <span m="3412290">birthday</span> <span m="3412870">is</span> <span m="3413020">October</span> <span m="3413820">28.</span> </p>
<p><span m="3415950">So</span> <span m="3416080">let's</span> <span m="3416310">go</span> <span m="3416580">right</span> <span m="3416740">across.</span> <span m="3417180">What</span> <span m="3417340">yours?</span> </p>
<p><span m="3417830">AUDIENCE:</span> <span m="3418260">April</span> <span m="3418690">1.</span> </p>
<p><span m="3419120">TOM LEIGHTON: April</span> <span m="3419860">1.</span> <span m="3426610">OK.</span> <span m="3427260">We</span> <span m="3427460">won't</span> <span m="3428200">embarrass</span> <span m="3428590">you</span> <span m="3428670">here.</span> <span m="3428810">OK,</span> <span m="3429000">who's</span> <span m="3429240">next?</span> <span m="3430930">What's</span> <span m="3431110">your</span> <span m="3431270">birthday?</span> </p>
<p><span m="3432283">AUDIENCE:</span> <span m="3432746">I'm</span> <span m="3433209">sorry.</span> <span m="3433672">September</span> <span m="3434135">2.</span> </p>
<p><span m="3434600">TOM LEIGHTON: September</span> <span m="3435260">2.</span> <span m="3435910">All</span> <span m="3436313">right.</span> <span m="3438020">Yours.</span> </p>
<p><span m="3438975">AUDIENCE:</span> <span m="3439290">June</span> <span m="3439340">1.</span> </p>
<p><span m="3439615">TOM LEIGHTON:</span> <span m="3439890">June</span> <span m="3440670">1.</span> <span m="3442440">OK.</span> <span m="3442880">We'll</span> <span m="3443170">come</span> <span m="3443360">back.</span> </p>
<p><span m="3443610">AUDIENCE:</span> <span m="3443750">April</span> <span m="3444030">8.</span> </p>
<p><span m="3444310">TOM LEIGHTON: What</span> <span m="3444440">is</span> <span m="3444540">it?</span> </p>
<p><span m="3444940">AUDIENCE: April</span> <span m="3445410">8.</span> </p>
<p><span m="3445880">TOM LEIGHTON: April</span> <span m="3446440">8.</span> <span m="3447481">All</span> <span m="3447972">right.</span> </p>
<p><span m="3448954">AUDIENCE:</span> <span m="3449445">November</span> <span m="3449936">20.</span> </p>
<p><span m="3450427">TOM LEIGHTON:</span> <span m="3450920">November</span> <span m="3451510">20.</span> </p>
<p><span m="3452992">AUDIENCE:</span> <span m="3453443">June</span> <span m="3453894">12.</span> </p>
<p><span m="3454796">TOM LEIGHTON:</span> <span m="3455250">June</span> <span m="3456260">12.</span> </p>
<p><span m="3457736">AUDIENCE:</span> <span m="3458212">December</span> <span m="3458688">29.</span> </p>
<p><span m="3459640">TOM LEIGHTON:</span> <span m="3460120">December</span> <span m="3460450">29.</span> </p>
<p><span m="3461952">AUDIENCE:</span> <span m="3462394">[INAUDIBLE].</span> </p>
<p><span m="3464162">TOM LEIGHTON:</span> <span m="3464610">What</span> <span m="3464750">is</span> <span m="3464850">it?</span> </p>
<p><span m="3465260">AUDIENCE: June</span> <span m="3465600">14.</span> </p>
<p><span m="3465940">TOM LEIGHTON:</span> <span m="3466280">June</span> <span m="3466880">14.</span> <span m="3467480">Ooh,</span> <span m="3467655">I</span> <span m="3467830">almost</span> <span m="3468240">got</span> <span m="3468450">one</span> <span m="3468590">there.</span> <span m="3468820">That</span> <span m="3468910">one's</span> <span m="3469070">close.</span> <span m="3470045">All</span> <span m="3470450">right.</span> <span m="3471260">What's</span> <span m="3471410">yours?</span> </p>
<p><span m="3471770">AUDIENCE:</span> <span m="3472266">March</span> <span m="3472762">6.</span> </p>
<p><span m="3473260">TOM LEIGHTON: March</span> <span m="3473650">6.</span> </p>
<p><span m="3475000">AUDIENCE:</span> <span m="3475450">May</span> <span m="3475897">2.</span> </p>
<p><span m="3476344">TOM LEIGHTON:</span> <span m="3476791">May</span> <span m="3477240">2.</span> </p>
<p><span m="3478650">AUDIENCE:</span> <span m="3479120">17th</span> <span m="3479590">of</span> <span m="3480060">November.</span> </p>
<p><span m="3480530">TOM LEIGHTON:</span> <span m="3481000">November</span> <span m="3481290">17.</span> <span m="3481580">Close</span> <span m="3481950">again.</span> </p>
<p><span m="3482530">AUDIENCE:</span> <span m="3482963">August</span> <span m="3483396">4.</span> </p>
<p><span m="3484695">TOM LEIGHTON:</span> <span m="3485130">August</span> <span m="3485840">4.</span> </p>
<p><span m="3487028">AUDIENCE:</span> <span m="3487516">July</span> <span m="3488004">25.</span> </p>
<p><span m="3488980">TOM LEIGHTON:</span> <span m="3489470">July</span> <span m="3489910">25.</span> <span m="3490785">I</span> <span m="3491230">don't</span> <span m="3491675">think</span> <span m="3492120">we'll</span> <span m="3492320">get</span> <span m="3492460">to</span> <span m="3492540">100</span> <span m="3492920">here,</span> <span m="3493280">hopefully.</span> <span m="3494170">Yeah,</span> <span m="3494660">what's</span> <span m="3494940">yours?</span> </p>
<p><span m="3495220">AUDIENCE:</span> <span m="3495635">October</span> <span m="3495842">30.</span> </p>
<p><span m="3496050">TOM LEIGHTON: What</span> <span m="3496210">is</span> <span m="3496300">it?</span> </p>
<p><span m="3496795">AUDIENCE: October</span> <span m="3497290">30.</span> </p>
<p><span m="3497710">TOM LEIGHTON: October</span> <span m="3498110">30.</span> <span m="3498510">Got</span> <span m="3498680">close.</span> </p>
<p><span m="3500716">AUDIENCE:</span> <span m="3501148">July</span> <span m="3501580">6.</span> </p>
<p><span m="3502012">TOM LEIGHTON:</span> <span m="3502450">July</span> <span m="3502890">6.</span> <span m="3503320">All</span> <span m="3503600">right.</span> </p>
<p><span m="3504726">AUDIENCE:</span> <span m="3505222">February</span> <span m="3505718">25.</span> </p>
<p><span m="3506214">TOM LEIGHTON:</span> <span m="3506710">February</span> <span m="3507080">25.</span> </p>
<p><span m="3510580">AUDIENCE:</span> <span m="3510992">May</span> <span m="3511404">21.</span> </p>
<p><span m="3512228">TOM LEIGHTON:</span> <span m="3512640">May</span> <span m="3512930">what?</span> <span m="3513930">21st</span> <span m="3514410">of</span> <span m="3514890">May.</span> </p>
<p><span m="3517630">AUDIENCE:</span> <span m="3517960">May</span> <span m="3518433">30.</span> </p>
<p><span m="3518906">TOM LEIGHTON:</span> <span m="3519380">May</span> <span m="3519900">30.</span> <span m="3521820">You</span> <span m="3522150">guys</span> <span m="3523110">fooled</span> <span m="3523370">me.</span> <span m="3523630">What</span> <span m="3523880">have</span> <span m="3524060">you</span> <span m="3524360">got?</span> </p>
<p><span m="3524660">AUDIENCE:</span> <span m="3525045">January</span> <span m="3525430">12.</span> </p>
<p><span m="3525815">TOM LEIGHTON:</span> <span m="3526200">January</span> <span m="3526870">12.</span> <span m="3527890">All</span> <span m="3527930">right.</span> </p>
<p><span m="3528852">AUDIENCE:</span> <span m="3529313">July</span> <span m="3529774">14.</span> </p>
<p><span m="3530696">TOM LEIGHTON:</span> <span m="3531160">July</span> <span m="3531910">14.</span> <span m="3535090">OK.</span> </p>
<p><span m="3535455">AUDIENCE:</span> <span m="3535820">April</span> <span m="3536265">30.</span> </p>
<p><span m="3537155">TOM LEIGHTON:</span> <span m="3537600">April</span> <span m="3538290">30.</span> </p>
<p><span m="3540303">AUDIENCE:</span> <span m="3540744">March</span> <span m="3541185">13.</span> </p>
<p><span m="3542067">TOM LEIGHTON:</span> <span m="3542510">March</span> <span m="3542960">13.</span> <span m="3545360">All</span> <span m="3545850">right.</span> <span m="3546000">Did</span> <span m="3546230">I</span> <span m="3546270">get--</span> </p>
<p><span m="3546610">AUDIENCE: October</span> <span m="3547090">7.</span> </p>
<p><span m="3547705">TOM LEIGHTON:</span> <span m="3548120">October</span> <span m="3548610">7.</span> </p>
<p><span m="3550044">AUDIENCE:</span> <span m="3550516">October</span> <span m="3550988">8.</span> </p>
<p><span m="3551460">TOM LEIGHTON:</span> <span m="3551932">Ah,</span> <span m="3552410">you</span> <span m="3552690">guys.</span> <span m="3556376">OK.</span> <span m="3557750">Did</span> <span m="3557820">I</span> <span m="3557880">get</span> <span m="3558060">you?</span> </p>
<p><span m="3558470">AUDIENCE:</span> <span m="3558893">September</span> <span m="3559316">15.</span> </p>
<p><span m="3559740">TOM LEIGHTON: September</span> <span m="3560460">15.</span> </p>
<p><span m="3562581">AUDIENCE:</span> <span m="3563048">November</span> <span m="3563515">9.</span> </p>
<p><span m="3564916">TOM LEIGHTON:</span> <span m="3565390">November</span> <span m="3565620">9.</span> <span m="3566464">All</span> <span m="3566886">right.</span> </p>
<p><span m="3567308">AUDIENCE: July</span> <span m="3567519">15.</span> </p>
<p><span m="3567730">TOM LEIGHTON: July</span> <span m="3569880">15.</span> <span m="3572190">Close.</span> </p>
<p><span m="3573306">AUDIENCE:</span> <span m="3573742">September</span> <span m="3574178">3.</span> </p>
<p><span m="3574614">TOM LEIGHTON:</span> <span m="3575050">September</span> <span m="3575650">3.</span> <span m="3576280">You</span> <span m="3576756">guys</span> <span m="3577232">are</span> <span m="3577710">killing</span> <span m="3577770">me</span> <span m="3578170">here.</span> </p>
<p><span m="3578680">AUDIENCE:</span> <span m="3579172">February</span> <span m="3579664">6.</span> </p>
<p><span m="3580156">TOM LEIGHTON:</span> <span m="3580650">February</span> <span m="3581310">6.</span> </p>
<p><span m="3581970">AUDIENCE:</span> <span m="3582434">October</span> <span m="3582898">26.</span> </p>
<p><span m="3584754">TOM LEIGHTON:</span> <span m="3585220">OK.</span> </p>
<p><span m="3586514">AUDIENCE:</span> <span m="3586978">November</span> <span m="3587442">2.</span> </p>
<p><span m="3588834">TOM LEIGHTON:</span> <span m="3589300">November</span> <span m="3589990">2.</span> </p>
<p><span m="3591163">AUDIENCE:</span> <span m="3591656">January</span> <span m="3592149">23.</span> </p>
<p><span m="3594121">TOM LEIGHTON:</span> <span m="3594620">January</span> <span m="3594885">23.</span> </p>
<p><span m="3596578">AUDIENCE:</span> <span m="3597054">September</span> <span m="3597530">27.</span> </p>
<p><span m="3599434">TOM LEIGHTON:</span> <span m="3599910">You</span> <span m="3600030">guys</span> <span m="3600240">are</span> <span m="3600280">going</span> <span m="3600360">to</span> <span m="3600420">set</span> <span m="3600650">a</span> <span m="3600700">record</span> <span m="3601060">for</span> <span m="3601160">sure</span> <span m="3601510">here.</span> <span m="3602230">This</span> <span m="3602560">isn't</span> <span m="3602700">the</span> <span m="3603110">way</span> <span m="3603156">it's</span> <span m="3603203">supposed</span> <span m="3603250">to</span> <span m="3603340">go.</span> </p>
<p><span m="3603890">AUDIENCE:</span> <span m="3604175">December</span> <span m="3604460">30.</span> </p>
<p><span m="3605292">TOM LEIGHTON:</span> <span m="3605710">December</span> <span m="3606260">30.</span> </p>
<p><span m="3609290">AUDIENCE:</span> <span m="3609790">December</span> <span m="3610235">28.</span> </p>
<p><span m="3610680">TOM LEIGHTON:</span> <span m="3611139">Ah,</span> <span m="3611598">come</span> <span m="3612057">on,</span> <span m="3612516">guys.</span> <span m="3615730">What</span> <span m="3615880">is</span> <span m="3616120">the</span> <span m="3616430">probability</span> <span m="3616690">of</span> <span m="3616950">going</span> <span m="3617210">this</span> <span m="3617370">long</span> <span m="3617660">here?</span> <span m="3618410">Yeah.</span> </p>
<p><span m="3619122">AUDIENCE:</span> <span m="3619554">September</span> <span m="3619986">22.</span> </p>
<p><span m="3620420">TOM LEIGHTON: September</span> <span m="3620910">22.</span> </p>
<p><span m="3623130">AUDIENCE:</span> <span m="3623597">July</span> <span m="3624064">30.</span> </p>
<p><span m="3626399">TOM LEIGHTON:</span> <span m="3626870">July</span> <span m="3627155">30.</span> </p>
<p><span m="3629396">AUDIENCE:</span> <span m="3629885">The</span> <span m="3630374">24th</span> <span m="3630863">of</span> <span m="3631352">August.</span> </p>
<p><span m="3632330">TOM LEIGHTON:</span> <span m="3632820">24th</span> <span m="3633200">August.</span> <span m="3633740">I'm</span> <span m="3633840">going</span> <span m="3633910">to</span> <span m="3633980">have</span> <span m="3634050">to</span> <span m="3634120">ask</span> <span m="3634310">the</span> <span m="3634410">same</span> <span m="3634670">person</span> <span m="3634980">to</span> <span m="3635050">tell</span> <span m="3635210">me</span> <span m="3635340">twice</span> <span m="3635710">here</span> <span m="3635850">to</span> <span m="3635900">get</span> <span m="3636030">a</span> <span m="3636080">match.</span> <span m="3637110">We</span> <span m="3637280">got</span> <span m="3637420">over</span> <span m="3637540">there</span> <span m="3637730">now?</span> </p>
<p><span m="3638150">AUDIENCE:</span> <span m="3638586">April</span> <span m="3639022">6.</span> </p>
<p><span m="3639460">TOM LEIGHTON: April</span> <span m="3640300">6.</span> </p>
<p><span m="3642141">AUDIENCE:</span> <span m="3642592">October</span> <span m="3643043">16.</span> </p>
<p><span m="3644396">TOM LEIGHTON:</span> <span m="3644850">October</span> <span m="3645110">16.</span> </p>
<p><span m="3645588">AUDIENCE:</span> <span m="3646066">Did</span> <span m="3646544">ask</span> <span m="3647022">how</span> <span m="3647261">many--</span> </p>
<p><span m="3647500">AUDIENCE: September</span> <span m="3647976">3.</span> </p>
<p><span m="3648452">TOM LEIGHTON: September</span> <span m="3648928">3.</span> <span m="3650460">All</span> <span m="3650510">right.</span> <span m="3652545">Very</span> <span m="3653455">good.</span> <span m="3655275">All</span> <span m="3655730">right.</span> <span m="3656190">Let's</span> <span m="3656495">count</span> <span m="3656800">and</span> <span m="3656920">see</span> <span m="3657290">how</span> <span m="3657610">many</span> <span m="3658670">we</span> <span m="3658860">got</span> <span m="3659150">here.</span> <span m="3660230">1,</span> <span m="3660677">2,</span> <span m="3660826">3,</span> <span m="3660975">4,</span> <span m="3661124">5,</span> <span m="3661571">6,</span> <span m="3662018">7,</span> <span m="3662465">8.</span> <span m="3662912">9,</span> <span m="3663359">10,</span> <span m="3663806">11,</span> <span m="3664260">12,</span> <span m="3664730">13,</span> <span m="3665200">14,</span> <span m="3665670">15,</span> <span m="3666140">16,</span> <span m="3667080">17,</span> <span m="3667550">18,</span> <span m="3668020">19,</span> <span m="3668490">20</span> <span m="3668960">21,</span> <span m="3669440">22,</span> <span m="3669892">23,</span> <span m="3671760">24,</span> <span m="3672060">25,</span> <span m="3672560">26,</span> <span m="3673060">27,</span> <span m="3674060">28,</span> <span m="3674560">29,</span> <span m="3675560">30,</span> <span m="3676060">31,</span> <span m="3676560">32,</span> <span m="3677060">33,</span> <span m="3677560">34,</span> <span m="3678060">35,</span> <span m="3678560">36,</span> <span m="3679060">37,</span> <span m="3679560">38,</span> <span m="3680060">39,</span> <span m="3681060">40,</span> <span m="3681560">41,</span> <span m="3681800">42.</span> <span m="3682630">That</span> <span m="3682890">is</span> <span m="3683070">a</span> <span m="3683160">record.</span> <span m="3684360">So</span> <span m="3684580">it</span> <span m="3684870">took</span> <span m="3684980">42</span> <span m="3685730">people</span> <span m="3687230">to</span> <span m="3687500">get</span> <span m="3687890">a</span> <span m="3687990">match.</span> </p>
<p><span m="3689230">Now</span> <span m="3689400">it</span> <span m="3689520">turns</span> <span m="3689880">out</span> <span m="3690180">that</span> <span m="3690260">for</span> <span m="3690530">N</span> <span m="3690740">equals</span> <span m="3691050">365,</span> <span m="3692750">the</span> <span m="3692860">magic</span> <span m="3693270">number</span> <span m="3693550">for</span> <span m="3693750">N</span> <span m="3694030">is</span> <span m="3694160">23,</span> <span m="3695090">that</span> <span m="3695520">by</span> <span m="3695660">23</span> <span m="3696420">people,</span> <span m="3696840">we</span> <span m="3696950">got</span> <span m="3697110">a</span> <span m="3697160">50-50</span> <span m="3697900">chance.</span> <span m="3699180">In</span> <span m="3699280">fact,</span> <span m="3699370">the</span> <span m="3699450">probability</span> <span m="3699940">of</span> <span m="3700020">a</span> <span m="3700070">match</span> <span m="3701080">on</span> <span m="3701270">23</span> <span m="3701800">people</span> <span m="3702180">is</span> <span m="3702320">0.506.</span> <span m="3704260">It's</span> <span m="3704580">a</span> <span m="3704660">little</span> <span m="3704860">bit</span> <span m="3705000">better</span> <span m="3705180">than</span> <span m="3705340">50-50</span> <span m="3705930">chance</span> <span m="3706400">at</span> <span m="3706640">23.</span> <span m="3707990">Now,</span> <span m="3708090">maybe</span> <span m="3708200">we</span> <span m="3708290">should</span> <span m="3708470">figure</span> <span m="3708720">out.</span> <span m="3708870">It's</span> <span m="3708912">too</span> <span m="3708955">late</span> <span m="3708997">for</span> <span m="3709040">homework</span> <span m="3709460">to</span> <span m="3709650">figure</span> <span m="3709900">out</span> <span m="3709980">what</span> <span m="3710110">the</span> <span m="3710200">chances</span> <span m="3710620">are</span> <span m="3710735">of</span> <span m="3710850">going</span> <span m="3711190">this</span> <span m="3711500">long</span> <span m="3712600">without</span> <span m="3712890">a</span> <span m="3712930">match.</span> <span m="3713340">That</span> <span m="3713760">maybe</span> <span m="3714000">worth</span> <span m="3714270">figuring</span> <span m="3714670">that</span> <span m="3714820">out.</span> </p>
<p><span m="3715910">Now,</span> <span m="3717390">it</span> <span m="3717480">may</span> <span m="3717690">seem</span> <span m="3717890">surprising</span> <span m="3718430">at</span> <span m="3718530">first</span> <span m="3718830">that</span> <span m="3718940">23</span> <span m="3719490">people</span> <span m="3719890">is</span> <span m="3719980">enough</span> <span m="3720500">to</span> <span m="3720640">have</span> <span m="3720820">a</span> <span m="3720900">50/50</span> <span m="3721530">chance</span> <span m="3722730">because</span> <span m="3722900">the</span> <span m="3722990">chance</span> <span m="3723350">of</span> <span m="3723430">any</span> <span m="3723650">pair</span> <span m="3724040">matching</span> <span m="3724470">is</span> <span m="3724580">1</span> <span m="3724790">in</span> <span m="3724880">365,</span> <span m="3726000">by</span> <span m="3726160">our</span> <span m="3726290">assumption.</span> <span m="3727770">And</span> <span m="3727920">that's</span> <span m="3728150">small,</span> <span m="3729230">but</span> <span m="3729840">there's</span> <span m="3730020">lots</span> <span m="3730260">of</span> <span m="3730350">pairs</span> <span m="3730670">of</span> <span m="3730760">people</span> <span m="3732330">and</span> <span m="3732570">every</span> <span m="3732800">pair</span> <span m="3733210">of</span> <span m="3733270">people</span> <span m="3733630">have</span> <span m="3733870">a</span> <span m="3733960">chance</span> <span m="3734630">to</span> <span m="3735100">match</span> <span m="3735950">and</span> <span m="3736080">that's</span> <span m="3736300">why</span> <span m="3736760">23</span> <span m="3737860">turns</span> <span m="3738170">out</span> <span m="3738310">to</span> <span m="3738360">be</span> <span m="3738450">enough</span> <span m="3738640">to</span> <span m="3738740">get</span> <span m="3738910">to</span> <span m="3738970">50-50.</span> </p>
<p><span m="3740420">Now,</span> <span m="3740580">we're</span> <span m="3740710">going</span> <span m="3740810">to</span> <span m="3740860">do</span> <span m="3741000">the</span> <span m="3741160">analysis</span> <span m="3742020">for</span> <span m="3742170">general</span> <span m="3742670">M</span> <span m="3743140">and</span> <span m="3743640">N</span> <span m="3743790">to</span> <span m="3743940">the</span> <span m="3744030">figure</span> <span m="3744330">out</span> <span m="3744470">the</span> <span m="3744550">probability</span> <span m="3745370">of</span> <span m="3745480">a</span> <span m="3745550">match</span> <span m="3746490">if</span> <span m="3746680">there's</span> <span m="3747850">M</span> <span m="3748160">people</span> <span m="3748820">and</span> <span m="3749050">N</span> <span m="3749580">birthdays.</span> <span m="3751230">There's</span> <span m="3751500">lots</span> <span m="3751870">of</span> <span m="3751960">ways</span> <span m="3752250">to</span> <span m="3752360">do</span> <span m="3752590">it.</span> <span m="3756140">The</span> <span m="3757260">easiest</span> <span m="3758140">is</span> <span m="3758300">to</span> <span m="3758400">sort</span> <span m="3758660">of</span> <span m="3759690">well,</span> <span m="3759900">we'll</span> <span m="3759980">draw</span> <span m="3760350">the</span> <span m="3760470">sample</span> <span m="3760870">space.</span> <span m="3761220">It</span> <span m="3761290">will</span> <span m="3761360">be</span> <span m="3761480">too</span> <span m="3761630">big</span> <span m="3762100">to</span> <span m="3762400">draw</span> <span m="3762620">the</span> <span m="3762710">whole</span> <span m="3762950">thing,</span> <span m="3763110">but</span> <span m="3763200">we</span> <span m="3763300">can</span> <span m="3763430">sort</span> <span m="3763590">of</span> <span m="3763670">model</span> <span m="3764020">the</span> <span m="3764130">sample</span> <span m="3764510">space</span> <span m="3766120">and</span> <span m="3766270">then</span> <span m="3766510">look</span> <span m="3766700">at</span> <span m="3766760">the</span> <span m="3766840">sample</span> <span m="3767050">points.</span> </p>
<p><span m="3781580">So</span> <span m="3782620">you've</span> <span m="3782790">got</span> <span m="3783060">the</span> <span m="3783330">first</span> <span m="3783660">person</span> <span m="3785680">and</span> <span m="3786030">there's</span> <span m="3786910">N</span> <span m="3787190">birthdays</span> <span m="3788400">here,</span> <span m="3788710">so</span> <span m="3788950">it</span> <span m="3789010">could</span> <span m="3789120">be</span> <span m="3789220">anywhere</span> <span m="3789430">from</span> <span m="3789560">January</span> <span m="3790060">1</span> <span m="3791250">out</span> <span m="3791600">to</span> <span m="3791960">December</span> <span m="3792320">31</span> <span m="3794630">and</span> <span m="3794730">in</span> <span m="3794820">general</span> <span m="3795320">this</span> <span m="3795450">will</span> <span m="3795580">be</span> <span m="3795810">N.</span> <span m="3798170">And</span> <span m="3798280">then</span> <span m="3798410">you</span> <span m="3798480">have</span> <span m="3798680">the</span> <span m="3798770">second</span> <span m="3799150">person</span> <span m="3807045">and</span> <span m="3807540">they</span> <span m="3808035">have</span> <span m="3808282">N</span> <span m="3808530">possibilities</span> <span m="3809280">for</span> <span m="3809390">their</span> <span m="3810450">birthday.</span> <span m="3811510">And</span> <span m="3812360">you</span> <span m="3812520">take</span> <span m="3812760">the</span> <span m="3812840">tree</span> <span m="3813210">down</span> <span m="3813520">M</span> <span m="3813820">levels</span> <span m="3817560">to</span> <span m="3817640">the</span> <span m="3817710">very</span> <span m="3817940">last</span> <span m="3818310">person</span> <span m="3818660">here.</span> </p>
<p><span m="3825920">So</span> <span m="3826840">each</span> <span m="3827110">node</span> <span m="3827390">has</span> <span m="3827590">degree</span> <span m="3827910">N</span> <span m="3828360">and</span> <span m="3828530">there's</span> <span m="3828820">M</span> <span m="3829020">levels</span> <span m="3829370">on</span> <span m="3829460">this</span> <span m="3829570">tree.</span> <span m="3830530">So</span> <span m="3830690">the</span> <span m="3830790">sample</span> <span m="3831210">space</span> <span m="3834710">is</span> <span m="3834940">the</span> <span m="3835040">set</span> <span m="3835510">of</span> <span m="3836720">all</span> <span m="3837870">n-tuples</span> <span m="3839280">b1,</span> <span m="3840700">b2,</span> <span m="3842233">to</span> <span m="3842674">bm,</span> <span m="3843560">these</span> <span m="3843810">are</span> <span m="3843950">the</span> <span m="3844620">birthdays</span> <span m="3846530">where</span> <span m="3848370">every</span> <span m="3848670">value</span> <span m="3849160">of</span> <span m="3849250">bi</span> <span m="3850880">is</span> <span m="3851040">between</span> <span m="3851636">1</span> <span m="3852092">and</span> <span m="3852548">N.</span> <span m="3855420">So</span> <span m="3855860">a</span> <span m="3855920">sample</span> <span m="3856330">point</span> <span m="3856670">is</span> <span m="3856780">all</span> <span m="3856930">the</span> <span m="3857010">birthdays</span> <span m="3857270">of</span> <span m="3857530">the</span> <span m="3857995">M</span> <span m="3858460">people.</span> </p>
<p><span m="3862180">How</span> <span m="3862370">many</span> <span m="3862580">sample</span> <span m="3862960">points</span> <span m="3863260">are</span> <span m="3863390">there</span> <span m="3863610">here?</span> <span m="3868460">Remember</span> <span m="3869720">how</span> <span m="3869830">to</span> <span m="3869920">count</span> <span m="3870200">these</span> <span m="3870420">things?</span> <span m="3872610">Number</span> <span m="3872840">of</span> <span m="3872920">leaves</span> <span m="3873290">on</span> <span m="3873420">an</span> <span m="3873580">N-ary</span> <span m="3873980">tree</span> <span m="3874330">of</span> <span m="3874450">depth</span> <span m="3874740">M</span> <span m="3875640">or</span> <span m="3875840">you</span> <span m="3875930">can</span> <span m="3876060">think</span> <span m="3876230">of</span> <span m="3876300">it</span> <span m="3876400">this</span> <span m="3876590">way.</span> <span m="3876780">I've</span> <span m="3876910">got</span> <span m="3878840">N</span> <span m="3879090">choices</span> <span m="3879580">for</span> <span m="3879725">each</span> <span m="3879870">bi</span> <span m="3880140">and</span> <span m="3880620">there's</span> <span m="3880980">M</span> <span m="3881330">of</span> <span m="3881480">them.</span> </p>
<p><span m="3881950">AUDIENCE:</span> <span m="3882347">[INAUDIBLE].</span> </p>
<p><span m="3883141">TOM LEIGHTON:</span> <span m="3883540">So</span> <span m="3883700">what's</span> <span m="3883900">the</span> <span m="3883990">number</span> <span m="3884290">of</span> <span m="3884400">sample</span> <span m="3884725">points?</span> </p>
<p><span m="3885050">AUDIENCE:</span> <span m="3885542">N</span> <span m="3886034">to</span> <span m="3886280">the</span> <span m="3886526">M.</span> </p>
<p><span m="3887020">TOM LEIGHTON:</span> <span m="3887435">N</span> <span m="3887850">to</span> <span m="3887960">the</span> <span m="3888330">M.</span> <span m="3891900">Because</span> <span m="3892660">N</span> <span m="3892960">choices</span> <span m="3893410">here,</span> <span m="3894720">N</span> <span m="3895000">choices</span> <span m="3895410">here,</span> <span m="3896780">N</span> <span m="3897020">choices</span> <span m="3897420">there,</span> <span m="3897660">so</span> <span m="3897740">you</span> <span m="3897820">have</span> <span m="3897900">N</span> <span m="3898150">times</span> <span m="3898420">N</span> <span m="3898640">times</span> <span m="3898920">N</span> <span m="3899360">M</span> <span m="3899800">times.</span> <span m="3902220">And</span> <span m="3903280">what's</span> <span m="3903590">the</span> <span m="3903690">probability</span> <span m="3904420">of</span> <span m="3904500">each</span> <span m="3904760">outcome?</span> <span m="3907160">For</span> <span m="3907300">a</span> <span m="3907360">set</span> <span m="3907590">of</span> <span m="3907670">possible</span> <span m="3908080">birthdays,</span> <span m="3908910">what's</span> <span m="3909200">its</span> <span m="3909320">probability?</span> <span m="3913460">What's</span> <span m="3913700">the</span> <span m="3913780">probability</span> <span m="3914870">of</span> <span m="3915960">b1,</span> <span m="3917180">b2,</span> <span m="3918920">bM?</span> </p>
<p><span m="3927460">So</span> <span m="3927560">the</span> <span m="3927630">probability</span> <span m="3928180">of</span> <span m="3928250">a</span> <span m="3928300">sample</span> <span m="3928650">point.</span> <span m="3928950">What's</span> <span m="3929120">the</span> <span m="3929180">probability</span> <span m="3929630">that</span> <span m="3929790">the</span> <span m="3929870">first</span> <span m="3930160">person</span> <span m="3930450">has</span> <span m="3930570">birthday</span> <span m="3930880">b1,</span> <span m="3931420">the</span> <span m="3931510">second</span> <span m="3931860">has</span> <span m="3931990">b2,</span> <span m="3932610">and</span> <span m="3932820">the</span> <span m="3932940">N-th</span> <span m="3933423">has</span> <span m="3933906">bM?</span> <span m="3936810">Remember</span> <span m="3936970">that?</span> <span m="3937618">Yeah.</span> </p>
<p><span m="3938046">AUDIENCE:</span> <span m="3938474">1</span> <span m="3938902">over</span> <span m="3938987">N</span> <span m="3939073">to</span> <span m="3939158">the</span> <span m="3939244">M.</span> </p>
<p><span m="3939330">TOM LEIGHTON:</span> <span m="3939760">1</span> <span m="3939980">over</span> <span m="3940170">N</span> <span m="3940470">to</span> <span m="3940553">the</span> <span m="3940636">M</span> <span m="3940720">because</span> <span m="3941030">each</span> <span m="3941410">edge</span> <span m="3942410">is</span> <span m="3942810">probability</span> <span m="3943035">of</span> <span m="3943260">1</span> <span m="3943510">over</span> <span m="3943760">N</span> <span m="3945700">and</span> <span m="3946080">the</span> <span m="3946180">paths</span> <span m="3946420">are</span> <span m="3946503">length</span> <span m="3946586">M,</span> <span m="3946670">so</span> <span m="3946780">you've</span> <span m="3946840">got</span> <span m="3947010">1</span> <span m="3947240">over</span> <span m="3947450">N</span> <span m="3947750">to</span> <span m="3947840">the</span> <span m="3947980">M-th</span> <span m="3948240">power.</span> <span m="3952770">Probability</span> <span m="3953025">of</span> <span m="3953280">the</span> <span m="3953370">first</span> <span m="3953680">birthday</span> <span m="3954050">matching</span> <span m="3954540">is</span> <span m="3954670">1</span> <span m="3954910">in</span> <span m="3955010">N</span> <span m="3955500">times</span> <span m="3955790">1</span> <span m="3956010">in</span> <span m="3956100">N</span> <span m="3957140">times</span> <span m="3957410">1</span> <span m="3957670">in</span> <span m="3957930">N.</span> <span m="3959840">And</span> <span m="3960020">this</span> <span m="3960080">actually</span> <span m="3960760">makes</span> <span m="3961060">sense</span> <span m="3961420">because</span> <span m="3963110">I've</span> <span m="3963240">got</span> <span m="3963490">N</span> <span m="3963720">to</span> <span m="3963770">the</span> <span m="3963930">M</span> <span m="3964070">sample</span> <span m="3964470">points,</span> <span m="3964890">each</span> <span m="3965150">a</span> <span m="3965220">probability</span> <span m="3965850">1</span> <span m="3966150">over</span> <span m="3966540">N</span> <span m="3966720">to</span> <span m="3966800">the</span> <span m="3966970">M.</span> <span m="3968000">So</span> <span m="3968140">they</span> <span m="3968300">all</span> <span m="3968500">add</span> <span m="3968610">up</span> <span m="3968720">to</span> <span m="3968820">1,</span> <span m="3970490">which</span> <span m="3970710">is</span> <span m="3970820">good.</span> </p>
<p><span m="3971920">What</span> <span m="3972170">kind</span> <span m="3972370">of</span> <span m="3972440">sample</span> <span m="3972810">space</span> <span m="3973140">is</span> <span m="3973260">this</span> <span m="3973550">where</span> <span m="3973650">this</span> <span m="3973820">happens</span> <span m="3974190">where</span> <span m="3974340">all</span> <span m="3974460">the</span> <span m="3974530">probabilities</span> <span m="3975040">are</span> <span m="3975070">the</span> <span m="3975160">same?</span> </p>
<p><span m="3975530">AUDIENCE:</span> <span m="3975916">Uniform.</span> </p>
<p><span m="3976302">TOM LEIGHTON:</span> <span m="3976690">Uniform.</span> <span m="3977580">Makes</span> <span m="3977780">it</span> <span m="3977870">very</span> <span m="3978210">easy</span> <span m="3978315">to</span> <span m="3978420">work</span> <span m="3978810">with.</span> <span m="3979080">All</span> <span m="3979130">we</span> <span m="3979180">got</span> <span m="3979230">to</span> <span m="3979310">do</span> <span m="3979470">now</span> <span m="3979770">is</span> <span m="3979880">just</span> <span m="3980470">count</span> <span m="3981240">the</span> <span m="3981330">number</span> <span m="3981550">of</span> <span m="3981610">sample</span> <span m="3981980">points</span> <span m="3982830">where</span> <span m="3982940">there's</span> <span m="3983110">a</span> <span m="3983160">matching</span> <span m="3983520">birthday</span> <span m="3985350">and</span> <span m="3985480">then</span> <span m="3985590">we</span> <span m="3985690">multiply</span> <span m="3986170">by</span> <span m="3986770">that</span> <span m="3986960">one</span> <span m="3987130">probability</span> <span m="3987560">1</span> <span m="3987770">over</span> <span m="3987950">N</span> <span m="3988100">to</span> <span m="3988180">the</span> <span m="3988320">M.</span> </p>
<p><span m="3990830">Now,</span> <span m="3991450">it</span> <span m="3991640">turns</span> <span m="3991990">out</span> <span m="3992280">that</span> <span m="3992640">rather</span> <span m="3992990">than</span> <span m="3993340">counting</span> <span m="3993780">the</span> <span m="3993870">number</span> <span m="3994360">of</span> <span m="3994440">sample</span> <span m="3994840">points</span> <span m="3995240">where</span> <span m="3995340">there's</span> <span m="3995500">a</span> <span m="3995550">matching</span> <span m="3996000">birthday,</span> <span m="3997180">it's</span> <span m="3997420">easier</span> <span m="3998300">to</span> <span m="3998490">count</span> <span m="3998920">the</span> <span m="3999040">number</span> <span m="3999310">of</span> <span m="3999380">sample</span> <span m="3999740">points</span> <span m="4000080">for</span> <span m="4000190">all</span> <span m="4000500">the</span> <span m="4000580">birthdays</span> <span m="4001290">are</span> <span m="4001430">different.</span> <span m="4003095">And</span> <span m="4003490">this</span> <span m="4003730">is</span> <span m="4003820">often</span> <span m="4004190">the</span> <span m="4004280">case</span> <span m="4004580">when</span> <span m="4004660">you're</span> <span m="4004750">doing</span> <span m="4004940">a</span> <span m="4005195">counting</span> <span m="4005450">problem,</span> <span m="4007470">it's</span> <span m="4008100">easier</span> <span m="4008460">to</span> <span m="4008560">count</span> <span m="4008900">the</span> <span m="4009040">opposite</span> <span m="4009740">of</span> <span m="4009940">what</span> <span m="4010220">you're</span> <span m="4010400">after.</span> <span m="4012100">That</span> <span m="4012300">can</span> <span m="4012450">be</span> <span m="4012640">the</span> <span m="4012760">case</span> <span m="4013150">and</span> <span m="4013280">it</span> <span m="4013340">is</span> <span m="4013530">the</span> <span m="4013610">case</span> <span m="4013900">here.</span> <span m="4014420">So</span> <span m="4014720">we're</span> <span m="4014760">going</span> <span m="4015010">to</span> <span m="4015260">do</span> <span m="4015760">that.</span> </p>
<p><span m="4019680">So</span> <span m="4022130">let's</span> <span m="4022480">count</span> <span m="4023240">how</span> <span m="4023540">many</span> <span m="4023780">sample</span> <span m="4024230">points</span> <span m="4025240">are</span> <span m="4025480">all</span> <span m="4026600">different</span> <span m="4027340">birthdays,</span> <span m="4028590">so</span> <span m="4028830">no</span> <span m="4029540">pair</span> <span m="4029890">of</span> <span m="4029980">bi's</span> <span m="4030410">is</span> <span m="4030640">the</span> <span m="4030740">same.</span> <span m="4032130">Let's</span> <span m="4032390">do</span> <span m="4032520">that.</span> <span m="4033180">How</span> <span m="4033360">many</span> <span m="4033590">choices</span> <span m="4034130">are</span> <span m="4034280">there</span> <span m="4034490">for</span> <span m="4034630">b1?</span> <span m="4035724">365</span> <span m="4038410">or</span> <span m="4038860">N.</span> <span m="4039540">Let's</span> <span m="4039700">do</span> <span m="4039760">this</span> <span m="4039820">in</span> <span m="4039890">terms</span> <span m="4040100">of</span> <span m="4040190">N</span> <span m="4040480">because</span> <span m="4040515">we're</span> <span m="4040550">going</span> <span m="4040640">to</span> <span m="4040680">use</span> <span m="4040930">this</span> <span m="4041110">for</span> <span m="4041200">general</span> <span m="4041530">N.</span> </p>
<p><span m="4042400">How</span> <span m="4042560">many</span> <span m="4042730">choices</span> <span m="4043140">for</span> <span m="4043270">b2?</span> <span m="4045470">N</span> <span m="4045720">minus</span> <span m="4046120">1.</span> <span m="4046360">Given</span> <span m="4046550">you</span> <span m="4046820">are</span> <span m="4046880">the</span> <span m="4046990">first</span> <span m="4047280">one,</span> <span m="4047340">you</span> <span m="4047460">can't</span> <span m="4047750">match</span> <span m="4048050">it.</span> <span m="4048950">And</span> <span m="4049060">then</span> <span m="4049220">N</span> <span m="4049390">minus</span> <span m="4049750">2</span> <span m="4052540">all</span> <span m="4052643">the</span> <span m="4052746">way</span> <span m="4052850">over</span> <span m="4052960">to</span> <span m="4053020">the</span> <span m="4053080">last</span> <span m="4053470">one</span> <span m="4053570">is</span> <span m="4053690">N</span> <span m="4053870">minus</span> <span m="4054230">M</span> <span m="4055260">plus</span> <span m="4055510">1.</span> <span m="4057360">And</span> <span m="4057690">this</span> <span m="4057890">is</span> <span m="4057990">a</span> <span m="4058050">formula</span> <span m="4058380">you</span> <span m="4058545">should</span> <span m="4058710">all</span> <span m="4059660">remember.</span> <span m="4061050">That's</span> <span m="4061230">just</span> <span m="4061660">N</span> <span m="4061880">factorial</span> <span m="4062560">over</span> <span m="4062790">N</span> <span m="4063010">minus</span> <span m="4063400">M</span> <span m="4064620">factorial.</span> <span m="4066430">You</span> <span m="4066590">did</span> <span m="4066790">this</span> <span m="4066940">sort</span> <span m="4067100">of</span> <span m="4067170">stuff</span> <span m="4068030">a</span> <span m="4068090">couple</span> <span m="4068350">weeks</span> <span m="4068580">ago</span> <span m="4068740">with</span> <span m="4068870">counting</span> <span m="4069300">sets</span> <span m="4069810">and</span> <span m="4069960">probability</span> <span m="4070490">is</span> <span m="4070620">really--</span> <span m="4071410">a</span> <span m="4071460">lot</span> <span m="4071640">of</span> <span m="4071710">it's</span> <span m="4071850">about</span> <span m="4072120">counting.</span> </p>
<p><span m="4074250">So</span> <span m="4074460">now</span> <span m="4074690">we</span> <span m="4074800">can</span> <span m="4074930">compute</span> <span m="4075550">the</span> <span m="4075660">probability</span> <span m="4076280">that</span> <span m="4076450">all</span> <span m="4076820">the</span> <span m="4076900">birthdays</span> <span m="4077450">are</span> <span m="4077500">different.</span> <span m="4084630">It's</span> <span m="4084880">just</span> <span m="4085230">adding</span> <span m="4085610">up</span> <span m="4087090">all</span> <span m="4087320">the</span> <span m="4087410">sample</span> <span m="4087750">points</span> <span m="4087990">of</span> <span m="4088230">which</span> <span m="4088450">there's</span> <span m="4088660">n</span> <span m="4088840">factorial</span> <span m="4089580">over</span> <span m="4090620">N</span> <span m="4090820">minus</span> <span m="4091240">M</span> <span m="4091410">factorial</span> <span m="4092870">and</span> <span m="4093500">multiply</span> <span m="4094080">by</span> <span m="4094210">the</span> <span m="4094320">probability</span> <span m="4094840">of</span> <span m="4094920">each</span> <span m="4095100">one,</span> <span m="4096310">which</span> <span m="4096529">is</span> <span m="4096630">1</span> <span m="4096870">over</span> <span m="4097090">N</span> <span m="4097290">to</span> <span m="4097350">the</span> <span m="4097529">M.</span> <span m="4102069">All</span> <span m="4102170">right.</span> <span m="4102430">So</span> <span m="4102740">we've</span> <span m="4102990">actually</span> <span m="4103260">now</span> <span m="4103490">answered</span> <span m="4104170">the</span> <span m="4104250">question.</span> <span m="4104630">This</span> <span m="4104800">is</span> <span m="4104880">the</span> <span m="4104979">probability</span> <span m="4106359">that</span> <span m="4106450">all</span> <span m="4106700">the</span> <span m="4106790">birthdays</span> <span m="4107229">are</span> <span m="4107290">different.</span> </p>
<p><span m="4108670">The</span> <span m="4108770">only</span> <span m="4108939">problem</span> <span m="4109300">is,</span> <span m="4109560">it's</span> <span m="4109720">not</span> <span m="4109939">so</span> <span m="4110149">clear</span> <span m="4112560">what</span> <span m="4112819">the</span> <span m="4112960">answer</span> <span m="4113359">is</span> <span m="4113580">to</span> <span m="4113760">actually</span> <span m="4114069">compute</span> <span m="4114479">this</span> <span m="4114720">or</span> <span m="4114790">how</span> <span m="4115270">fast</span> <span m="4115689">it</span> <span m="4115760">grows.</span> <span m="4117250">So</span> <span m="4117410">if</span> <span m="4117500">I</span> <span m="4117550">wanted</span> <span m="4117880">to</span> <span m="4117950">get</span> <span m="4118170">a</span> <span m="4119310">closed</span> <span m="4119760">form</span> <span m="4120109">for</span> <span m="4120210">this</span> <span m="4120410">without</span> <span m="4120640">the</span> <span m="4120720">factorials,</span> <span m="4122170">what</span> <span m="4122319">do</span> <span m="4122420">I</span> <span m="4122510">do?</span> <span m="4122979">What</span> <span m="4123109">do</span> <span m="4123189">I</span> <span m="4123270">use?</span> <span m="4125200">Stirling's</span> <span m="4125790">formula.</span> </p>
<p><span m="4128310">So</span> <span m="4128720">let's</span> <span m="4128859">remember</span> <span m="4129100">that.</span> <span m="4137630">It</span> <span m="4137770">says</span> <span m="4138090">that</span> <span m="4138200">N</span> <span m="4138370">factorial</span> <span m="4140200">is</span> <span m="4140680">asymptotically</span> <span m="4141510">equal</span> <span m="4141870">to</span> <span m="4142240">square</span> <span m="4142609">root</span> <span m="4142819">2</span> <span m="4143010">pi</span> <span m="4143340">N</span> <span m="4144990">times</span> <span m="4146010">N</span> <span m="4146229">over</span> <span m="4146520">e</span> <span m="4147779">to</span> <span m="4147850">the</span> <span m="4148000">N.</span> <span m="4149830">And</span> <span m="4150240">that</span> <span m="4150460">is</span> <span m="4150670">accurate</span> <span m="4151660">within</span> <span m="4152000">0.1%</span> <span m="4153680">when</span> <span m="4154229">N</span> <span m="4154430">is</span> <span m="4154550">at</span> <span m="4154630">least</span> <span m="4154850">100.</span> <span m="4155529">So</span> <span m="4155899">not</span> <span m="4155960">only</span> <span m="4156330">is</span> <span m="4156520">it</span> <span m="4156779">asymptotically</span> <span m="4157510">equal,</span> <span m="4158240">it's</span> <span m="4158550">right</span> <span m="4158960">on</span> <span m="4159149">track</span> <span m="4160200">for</span> <span m="4160300">a</span> <span m="4160370">reasonable</span> <span m="4160680">size</span> <span m="4161162">N.</span> </p>
<p><span m="4165020">Now,</span> <span m="4165600">I</span> <span m="4165740">won't</span> <span m="4166140">drag</span> <span m="4166500">you</span> <span m="4166590">through</span> <span m="4166810">all</span> <span m="4167109">the</span> <span m="4167200">calculations.</span> <span m="4167899">I</span> <span m="4168229">used</span> <span m="4168430">to</span> <span m="4168500">actually</span> <span m="4168790">try</span> <span m="4169859">plugging</span> <span m="4170240">that</span> <span m="4170470">formula</span> <span m="4170850">in</span> <span m="4171115">for</span> <span m="4171380">here</span> <span m="4171630">and</span> <span m="4171689">here</span> <span m="4172380">and</span> <span m="4172490">then</span> <span m="4172680">going</span> <span m="4173050">through</span> <span m="4173250">all</span> <span m="4173510">the</span> <span m="4173609">calculations,</span> <span m="4174410">but</span> <span m="4174430">we</span> <span m="4174529">won't</span> <span m="4174890">do</span> <span m="4175029">it</span> <span m="4175080">in</span> <span m="4175160">class.</span> <span m="4175440">It's</span> <span m="4175720">in</span> <span m="4175790">the</span> <span m="4175899">text.</span> <span m="4177300">But</span> <span m="4177350">I</span> <span m="4177399">will</span> <span m="4177560">tell</span> <span m="4177729">you</span> <span m="4177859">where</span> <span m="4177970">that</span> <span m="4178180">winds</span> <span m="4178510">up.</span> <span m="4180620">It's</span> <span m="4180790">not</span> <span m="4180930">hard,</span> <span m="4181210">you've</span> <span m="4181290">just</span> <span m="4181420">got</span> <span m="4181540">to</span> <span m="4181660">do</span> <span m="4181930">the</span> <span m="4182200">calculation.</span> </p>
<p><span m="4188160">So</span> <span m="4188330">this</span> <span m="4188580">is</span> <span m="4188790">means</span> <span m="4188815">the</span> <span m="4188840">probability</span> <span m="4189359">that</span> <span m="4190670">all</span> <span m="4190950">birthdays</span> <span m="4191380">are</span> <span m="4191430">different</span> <span m="4198350">turns</span> <span m="4198710">out</span> <span m="4198990">to</span> <span m="4199080">be</span> <span m="4200030">asymptotically</span> <span m="4201000">equal</span> <span m="4201350">to</span> <span m="4202820">e</span> <span m="4204830">to</span> <span m="4205180">the</span> <span m="4205880">N</span> <span m="4206130">minus</span> <span m="4206530">M</span> <span m="4207080">plus</span> <span m="4208510">1/2</span> <span m="4211100">times</span> <span m="4211700">the</span> <span m="4211810">natural</span> <span m="4212280">log</span> <span m="4214030">of</span> <span m="4214360">N</span> <span m="4215000">over</span> <span m="4215340">N</span> <span m="4215530">minus</span> <span m="4215910">M</span> <span m="4218300">minus</span> <span m="4219130">M.</span> <span m="4220890">And</span> <span m="4221060">that's</span> <span m="4221340">accurate</span> <span m="4221565">to</span> <span m="4221790">within</span> <span m="4222050">0.2%,</span> <span m="4223670">if</span> <span m="4223870">N</span> <span m="4224036">and</span> <span m="4224203">N</span> <span m="4224370">minus</span> <span m="4224700">M</span> <span m="4224840">are</span> <span m="4224900">large,</span> <span m="4225270">larger</span> <span m="4225510">than</span> <span m="4225590">100.</span> <span m="4225950">So</span> <span m="4226360">in</span> <span m="4226450">fact,</span> <span m="4226880">it's</span> <span m="4227310">almost</span> <span m="4228020">equal.</span> </p>
<p><span m="4229830">And</span> <span m="4229950">now</span> <span m="4230090">you</span> <span m="4230230">could</span> <span m="4230400">plug</span> <span m="4230880">in</span> <span m="4232220">N</span> <span m="4232440">equals</span> <span m="4232780">365</span> <span m="4233840">and</span> <span m="4233960">M</span> <span m="4234150">equals</span> <span m="4234390">100.</span> <span m="4236650">So</span> <span m="4236820">if</span> <span m="4236890">you</span> <span m="4236990">do</span> <span m="4237170">that,</span> <span m="4240490">in</span> <span m="4240690">fact,</span> <span m="4240930">if</span> <span m="4241010">somebody</span> <span m="4241090">has</span> <span m="4241280">a</span> <span m="4241340">calculator,</span> <span m="4241950">we</span> <span m="4242030">should</span> <span m="4242200">plug</span> <span m="4242525">in,</span> <span m="4242850">what</span> <span m="4243280">do</span> <span m="4243430">we</span> <span m="4243560">have,</span> <span m="4245600">42.</span> <span m="4245950">You</span> <span m="4246125">should</span> <span m="4246300">plug</span> <span m="4246480">in</span> <span m="4246570">M</span> <span m="4246660">equals</span> <span m="4247260">42</span> <span m="4247740">and</span> <span m="4247850">see</span> <span m="4248020">what</span> <span m="4248180">the</span> <span m="4248280">probability</span> <span m="4248770">is.</span> <span m="4250300">But</span> <span m="4250420">if</span> <span m="4250510">M</span> <span m="4250690">is</span> <span m="4250790">100,</span> <span m="4251990">the</span> <span m="4252090">chance</span> <span m="4252680">that</span> <span m="4253770">we're</span> <span m="4253980">all</span> <span m="4254230">different,</span> <span m="4256330">this</span> <span m="4256600">equals</span> <span m="4257220">3.07</span> <span m="4259320">dot,</span> <span m="4259470">dot,</span> <span m="4259640">dot</span> <span m="4260530">times</span> <span m="4260900">10</span> <span m="4261680">to</span> <span m="4262190">the</span> <span m="4262290">minus</span> <span m="4262700">7.</span> <span m="4265150">And</span> <span m="4265270">we</span> <span m="4265360">should</span> <span m="4265550">check</span> <span m="4265900">for</span> <span m="4266040">M</span> <span m="4266250">equals</span> <span m="4266540">42.</span> <span m="4267720">My</span> <span m="4267840">guess</span> <span m="4268080">is</span> <span m="4268170">it's</span> <span m="4268380">pretty</span> <span m="4268770">small,</span> <span m="4269510">but</span> <span m="4269630">I</span> <span m="4269950">don't</span> <span m="4270015">know.</span> <span m="4270080">We'll</span> <span m="4270145">have</span> <span m="4270210">to</span> <span m="4270320">check</span> <span m="4270580">that.</span> </p>
<p><span m="4271351">AUDIENCE:</span> <span m="4271822">0.0859.</span> </p>
<p><span m="4274648">TOM LEIGHTON:</span> <span m="4275120">Great.</span> <span m="4275540">So</span> <span m="4276670">a</span> <span m="4276750">9%</span> <span m="4277520">chance</span> <span m="4278460">of</span> <span m="4279390">having</span> <span m="4279630">42</span> <span m="4280100">people</span> <span m="4280365">all</span> <span m="4280630">miss</span> <span m="4281020">is</span> <span m="4281055">a</span> <span m="4281090">9%</span> <span m="4281660">chance.</span> <span m="4282190">So</span> <span m="4282240">we</span> <span m="4282340">were</span> <span m="4282470">little</span> <span m="4283070">unlucky.</span> <span m="4283480">That</span> <span m="4283650">won't</span> <span m="4283830">happen</span> <span m="4284170">very</span> <span m="4284610">often.</span> <span m="4285770">But</span> <span m="4285990">when</span> <span m="4286080">you</span> <span m="4286160">go</span> <span m="4286340">from</span> <span m="4286470">42</span> <span m="4286920">to</span> <span m="4287020">100,</span> <span m="4287430">it</span> <span m="4287490">gets</span> <span m="4287700">really</span> <span m="4288280">small.</span> <span m="4289720">1</span> <span m="4290070">in</span> <span m="4290545">3</span> <span m="4290663">million</span> <span m="4290782">or</span> <span m="4290901">so.</span> <span m="4293870">Now,</span> <span m="4294630">if</span> <span m="4294840">N</span> <span m="4295160">is</span> <span m="4295480">365</span> <span m="4296360">and</span> <span m="4296710">M</span> <span m="4296770">is</span> <span m="4296830">23,</span> <span m="4299160">the</span> <span m="4299270">probability</span> <span m="4299840">comes</span> <span m="4300110">out</span> <span m="4300310">to</span> <span m="4300390">be</span> <span m="4300540">about</span> <span m="4300970">0.49,</span> <span m="4303330">so</span> <span m="4303510">about</span> <span m="4303830">50-50,</span> <span m="4304190">they're</span> <span m="4305142">all</span> <span m="4305618">different.</span> </p>
<p><span m="4310810">Now.</span> <span m="4311040">For</span> <span m="4311220">general</span> <span m="4311730">M</span> <span m="4311896">and</span> <span m="4312063">N,</span> <span m="4312230">we'd</span> <span m="4312390">like</span> <span m="4312610">to</span> <span m="4312730">know</span> <span m="4314210">when</span> <span m="4314620">do</span> <span m="4314690">you</span> <span m="4314810">get</span> <span m="4314960">to</span> <span m="4315020">the</span> <span m="4315140">50-50</span> <span m="4315760">point?</span> <span m="4316640">We'd</span> <span m="4316730">like</span> <span m="4316850">to</span> <span m="4316940">derive</span> <span m="4317400">an</span> <span m="4317450">equation</span> <span m="4318190">for</span> <span m="4318430">M</span> <span m="4318880">in</span> <span m="4319050">terms</span> <span m="4319430">of</span> <span m="4319520">N</span> <span m="4320550">where</span> <span m="4320640">the</span> <span m="4320760">probability</span> <span m="4321370">of</span> <span m="4321420">being</span> <span m="4321580">all</span> <span m="4321770">different</span> <span m="4322120">is</span> <span m="4322240">about</span> <span m="4322530">1/2.</span> <span m="4324590">All</span> <span m="4324680">right.</span> <span m="4324850">So</span> <span m="4324940">let's</span> <span m="4325120">do</span> <span m="4325240">that.</span> <span m="4336270">So</span> <span m="4339460">as</span> <span m="4339590">long</span> <span m="4339810">as</span> <span m="4339900">we</span> <span m="4340160">assume--</span> <span m="4340780">and</span> <span m="4340880">this</span> <span m="4341040">will</span> <span m="4341280">turn</span> <span m="4341385">out</span> <span m="4341490">to</span> <span m="4341600">be</span> <span m="4341670">true--</span> <span m="4342500">that</span> <span m="4342640">M</span> <span m="4342890">is</span> <span m="4342950">a</span> <span m="4343010">little</span> <span m="4343400">o</span> <span m="4343600">of</span> <span m="4343750">N</span> <span m="4343910">to</span> <span m="4343960">the</span> <span m="4344060">2/3</span> <span m="4344830">and</span> <span m="4344960">remember</span> <span m="4345250">little</span> <span m="4345500">o</span> <span m="4345740">means</span> <span m="4345990">it</span> <span m="4346070">grows</span> <span m="4346360">slower</span> <span m="4347070">than</span> <span m="4347420">N</span> <span m="4347630">to</span> <span m="4347690">the</span> <span m="4347780">2/3.</span> <span m="4350190">Then</span> <span m="4350250">we</span> <span m="4350350">can</span> <span m="4350490">simplify</span> <span m="4351070">that</span> <span m="4351250">expression</span> <span m="4352230">in</span> <span m="4352390">asymptotic</span> <span m="4353030">notation.</span> </p>
<p><span m="4355240">And</span> <span m="4355390">when</span> <span m="4355520">you</span> <span m="4355620">do</span> <span m="4355900">it,</span> <span m="4358570">I</span> <span m="4358700">won't</span> <span m="4358990">drag</span> <span m="4359240">it</span> <span m="4359310">through</span> <span m="4359490">on</span> <span m="4359600">the</span> <span m="4359670">board.</span> <span m="4360405">It's</span> <span m="4360670">also</span> <span m="4360970">in</span> <span m="4361055">the</span> <span m="4361140">text,</span> <span m="4362510">it</span> <span m="4362870">turns</span> <span m="4363220">out</span> <span m="4363400">to</span> <span m="4363460">be</span> <span m="4363610">much</span> <span m="4363860">simpler.</span> <span m="4365030">It's</span> <span m="4365200">just</span> <span m="4365460">e</span> <span m="4365700">to</span> <span m="4365760">the</span> <span m="4365860">minus</span> <span m="4366450">M</span> <span m="4366650">squared</span> <span m="4367730">over</span> <span m="4367950">2N.</span> <span m="4369900">So</span> <span m="4370030">I</span> <span m="4370110">take</span> <span m="4370390">that</span> <span m="4370590">thing</span> <span m="4370770">up</span> <span m="4370910">there</span> <span m="4371130">and</span> <span m="4371230">I</span> <span m="4371820">assume</span> <span m="4372300">that</span> <span m="4372500">M</span> <span m="4372820">is</span> <span m="4373790">growing</span> <span m="4374500">less</span> <span m="4374810">fast</span> <span m="4375200">than</span> <span m="4375300">the</span> <span m="4375390">2/3</span> <span m="4375810">power</span> <span m="4376140">of</span> <span m="4376220">N</span> <span m="4377400">and</span> <span m="4377540">that</span> <span m="4377700">whole</span> <span m="4377990">upper</span> <span m="4378260">expression</span> <span m="4378770">reduces</span> <span m="4379250">down</span> <span m="4379500">to</span> <span m="4379590">M</span> <span m="4379790">squared</span> <span m="4380100">over</span> <span m="4380260">2N.</span> <span m="4381110">Everything</span> <span m="4381420">else</span> <span m="4381860">goes</span> <span m="4382120">to</span> <span m="4382210">0</span> <span m="4382970">in</span> <span m="4383080">the</span> <span m="4383160">exponent.</span> <span m="4384310">Doesn't</span> <span m="4384590">matter.</span> </p>
<p><span m="4386260">Now,</span> <span m="4386960">if</span> <span m="4387110">I</span> <span m="4387180">set</span> <span m="4387530">this</span> <span m="4387740">to</span> <span m="4387840">be</span> <span m="4388070">1/2,</span> <span m="4391670">I</span> <span m="4391810">can</span> <span m="4391970">solve</span> <span m="4392490">this</span> <span m="4392760">to</span> <span m="4392850">find</span> <span m="4393130">out</span> <span m="4393310">what</span> <span m="4393530">M</span> <span m="4393785">has</span> <span m="4394040">to</span> <span m="4394130">be</span> <span m="4394800">to</span> <span m="4394920">make</span> <span m="4395110">that</span> <span m="4395290">be</span> <span m="4395430">1/2.</span> <span m="4397740">All</span> <span m="4397870">right.</span> <span m="4398110">So</span> <span m="4398210">this</span> <span m="4398310">will</span> <span m="4398470">be</span> <span m="4398580">true</span> <span m="4400230">if</span> <span m="4400390">and</span> <span m="4400470">only</span> <span m="4400810">if</span> <span m="4401080">minus</span> <span m="4401640">M</span> <span m="4401830">squared</span> <span m="4402840">over</span> <span m="4403080">2N</span> <span m="4404710">is</span> <span m="4404870">equal</span> <span m="4405170">to</span> <span m="4405265">the</span> <span m="4405360">natural</span> <span m="4405800">log</span> <span m="4406140">of</span> <span m="4406220">1/2.</span> <span m="4409610">And</span> <span m="4409900">that's</span> <span m="4410200">true.</span> <span m="4411040">Take</span> <span m="4411220">the</span> <span m="4411290">minus</span> <span m="4411670">sign,</span> <span m="4412250">put</span> <span m="4412345">it</span> <span m="4412440">inside</span> <span m="4412700">to</span> <span m="4412960">make</span> <span m="4413045">a</span> <span m="4413130">log</span> <span m="4413490">of</span> <span m="4413590">2,</span> <span m="4414820">multiply</span> <span m="4415240">by</span> <span m="4415410">2N.</span> <span m="4416580">That's</span> <span m="4416850">true</span> <span m="4417100">if</span> <span m="4417200">M</span> <span m="4417250">squared</span> <span m="4417720">equals</span> <span m="4418310">2N</span> <span m="4420510">natural</span> <span m="4420920">log</span> <span m="4421180">of</span> <span m="4421290">2.</span> </p>
<p><span m="4423590">And</span> <span m="4423780">now</span> <span m="4424010">I</span> <span m="4424075">can</span> <span m="4424140">solve</span> <span m="4424520">for</span> <span m="4424650">M</span> <span m="4424880">really</span> <span m="4425200">easily.</span> <span m="4426800">That's</span> <span m="4427100">true</span> <span m="4428570">if</span> <span m="4428770">and</span> <span m="4428850">only</span> <span m="4429150">if</span> <span m="4430350">M</span> <span m="4431660">equals</span> <span m="4433090">the</span> <span m="4433280">square</span> <span m="4433790">root</span> <span m="4434410">of</span> <span m="4434550">2</span> <span m="4435210">natural</span> <span m="4435610">log</span> <span m="4435810">of</span> <span m="4435900">2N,</span> <span m="4438090">which</span> <span m="4438360">is</span> <span m="4438530">about</span> <span m="4439420">1.177</span> <span m="4441172">square</span> <span m="4441610">root</span> <span m="4442083">of</span> <span m="4442556">N.</span> <span m="4445870">So</span> <span m="4446060">for</span> <span m="4446200">general</span> <span m="4446790">N,</span> <span m="4448560">you</span> <span m="4448700">get</span> <span m="4448910">a</span> <span m="4448960">50%</span> <span m="4449720">probability</span> <span m="4451200">of</span> <span m="4451370">having</span> <span m="4451680">a</span> <span m="4451730">matching</span> <span m="4452250">birthday</span> <span m="4453710">when</span> <span m="4454020">M</span> <span m="4454440">is</span> <span m="4454620">in</span> <span m="4454720">this</span> <span m="4454920">range,</span> <span m="4457780">pretty</span> <span m="4458030">close</span> <span m="4458330">to</span> <span m="4458450">1.2</span> <span m="4459270">square</span> <span m="4459570">root</span> <span m="4459650">of</span> <span m="4459740">N.</span> </p>
<p><span m="4461920">Now,</span> <span m="4462360">this</span> <span m="4462660">square</span> <span m="4463080">root</span> <span m="4463290">N</span> <span m="4463670">phenomenon,</span> <span m="4464440">this</span> <span m="4464660">thing</span> <span m="4464850">here,</span> <span m="4465490">that's</span> <span m="4465740">what's</span> <span m="4465940">known</span> <span m="4466130">as</span> <span m="4466230">the</span> <span m="4466310">birthday</span> <span m="4466650">principle.</span> <span m="4469630">It</span> <span m="4469770">says</span> <span m="4470100">if</span> <span m="4470240">you've</span> <span m="4470350">got</span> <span m="4471490">roughly</span> <span m="4471900">square</span> <span m="4472260">root</span> <span m="4472360">of</span> <span m="4472460">N</span> <span m="4472670">randomly</span> <span m="4473250">allocated</span> <span m="4473800">items</span> <span m="4474890">into</span> <span m="4475390">N</span> <span m="4476070">boxes</span> <span m="4476810">or</span> <span m="4476930">bins</span> <span m="4477680">or</span> <span m="4477830">birthdays,</span> <span m="4479580">there's</span> <span m="4479780">a</span> <span m="4479860">decent</span> <span m="4480280">chance</span> <span m="4480930">two</span> <span m="4481170">of</span> <span m="4481410">the</span> <span m="4481600">items</span> <span m="4481940">will</span> <span m="4482060">go</span> <span m="4482260">into</span> <span m="4482380">the</span> <span m="4482490">same</span> <span m="4482820">bin</span> <span m="4484340">if</span> <span m="4484460">the</span> <span m="4484610">randomly</span> <span m="4485230">allocated.</span> <span m="4487120">In</span> <span m="4487370">this</span> <span m="4487570">case,</span> <span m="4487770">the</span> <span m="4487850">bins</span> <span m="4488240">are</span> <span m="4488280">the</span> <span m="4488380">possible</span> <span m="4488860">days</span> <span m="4489130">of</span> <span m="4489200">the</span> <span m="4489310">year</span> <span m="4489670">that</span> <span m="4489900">we</span> <span m="4490440">put</span> <span m="4490630">each</span> <span m="4490790">person</span> <span m="4491090">into</span> <span m="4491320">for</span> <span m="4491440">their</span> <span m="4491560">birthday.</span> <span m="4493150">Any</span> <span m="4493310">questions</span> <span m="4494555">about</span> <span m="4495040">that?</span> <span m="4498920">Yeah.</span> </p>
<p><span m="4499360">AUDIENCE:</span> <span m="4499852">M</span> <span m="4500098">and</span> <span m="4500344">N</span> <span m="4500508">are</span> <span m="4500672">like</span> <span m="4500836">numbers</span> <span m="4501328">like</span> <span m="4501820">they're</span> <span m="4501943">defined</span> <span m="4502066">up</span> <span m="4502189">there</span> <span m="4502312">or</span> <span m="4502804">does</span> <span m="4503050">it</span> <span m="4503296">mean</span> <span m="4503460">to</span> <span m="4503624">say</span> <span m="4503788">M</span> <span m="4504280">equals</span> <span m="4504772">[INAUDIBLE]?</span> </p>
<p><span m="4505756">TOM LEIGHTON:</span> <span m="4506260">Yeah.</span> <span m="4506980">So</span> <span m="4507850">here</span> <span m="4508160">I</span> <span m="4508240">looked</span> <span m="4508353">at</span> <span m="4508466">a</span> <span m="4508580">special</span> <span m="4509070">case</span> <span m="4509490">where</span> <span m="4509600">N</span> <span m="4509815">was</span> <span m="4510030">365,</span> <span m="4510370">M</span> <span m="4510710">was</span> <span m="4511203">100,</span> <span m="4512190">but</span> <span m="4512320">we</span> <span m="4512440">can</span> <span m="4512580">imagine</span> <span m="4513160">them</span> <span m="4513360">as</span> <span m="4513530">arbitrary</span> <span m="4514290">numbers</span> <span m="4515420">that</span> <span m="4515550">could</span> <span m="4515700">be</span> <span m="4515790">getting</span> <span m="4516030">large.</span> <span m="4517350">And</span> <span m="4517640">so</span> <span m="4517780">over</span> <span m="4518030">here</span> <span m="4518300">and</span> <span m="4518380">I</span> <span m="4518470">say</span> <span m="4519550">M</span> <span m="4519920">is</span> <span m="4520460">little</span> <span m="4520660">o</span> <span m="4520890">of</span> <span m="4520953">N</span> <span m="4521016">to</span> <span m="4521080">the</span> <span m="4521240">2/3,</span> <span m="4522480">I</span> <span m="4522690">mean,</span> <span m="4523996">well,</span> <span m="4524342">M</span> <span m="4524516">equals</span> <span m="4524690">square</span> <span m="4524970">root</span> <span m="4525090">of</span> <span m="4525180">N</span> <span m="4525390">would</span> <span m="4525520">qualify.</span> <span m="4526250">Square</span> <span m="4526620">root</span> <span m="4526720">of</span> <span m="4526830">N</span> <span m="4527060">is</span> <span m="4527350">little</span> <span m="4527590">o</span> <span m="4527665">of</span> <span m="4527740">N</span> <span m="4527820">to</span> <span m="4527900">the</span> <span m="4528050">2/3.</span> <span m="4529950">So</span> <span m="4530100">as</span> <span m="4530210">long</span> <span m="4530590">as</span> <span m="4531310">M</span> <span m="4531540">is</span> <span m="4531650">not</span> <span m="4531830">growing</span> <span m="4532080">too</span> <span m="4532250">fast,</span> <span m="4533320">I</span> <span m="4533410">can</span> <span m="4533570">simplify</span> <span m="4534260">that</span> <span m="4534450">expression</span> <span m="4534760">up</span> <span m="4535070">there,</span> <span m="4536270">which</span> <span m="4536420">is</span> <span m="4536490">what</span> <span m="4536650">I</span> <span m="4536720">did.</span> </p>
<p><span m="4536950">And</span> <span m="4537030">then</span> <span m="4537130">we</span> <span m="4537220">go</span> <span m="4537360">back</span> <span m="4537650">and</span> <span m="4537740">we</span> <span m="4537860">find,</span> <span m="4538860">in</span> <span m="4539290">fact,</span> <span m="4540890">the</span> <span m="4541090">square</span> <span m="4541320">root</span> <span m="4541450">of</span> <span m="4541700">N</span> <span m="4541920">the</span> <span m="4542000">right</span> <span m="4542180">answer</span> <span m="4542980">and</span> <span m="4543120">that</span> <span m="4543300">is</span> <span m="4543510">little</span> <span m="4543810">o</span> <span m="4543864">of</span> <span m="4543972">N</span> <span m="4544026">to</span> <span m="4544080">the</span> <span m="4544160">2/3.</span> <span m="4545650">And</span> <span m="4545770">I</span> <span m="4545820">have</span> <span m="4546010">to</span> <span m="4546070">use</span> <span m="4546250">a</span> <span m="4546290">different</span> <span m="4546840">argument</span> <span m="4548400">if</span> <span m="4548790">I</span> <span m="4548890">assumed</span> <span m="4549260">M</span> <span m="4549430">was</span> <span m="4549590">bigger,</span> <span m="4550380">which</span> <span m="4550570">I</span> <span m="4550620">didn't</span> <span m="4551030">do.</span> <span m="4551750">I</span> <span m="4551850">didn't</span> <span m="4551970">drag</span> <span m="4552230">it</span> <span m="4552340">for</span> <span m="4552450">that.</span> <span m="4553140">But</span> <span m="4553270">I</span> <span m="4553290">would</span> <span m="4553460">have</span> <span m="4553550">to</span> <span m="4553630">go</span> <span m="4553710">check</span> <span m="4553950">that</span> <span m="4554110">case.</span> </p>
<p><span m="4556340">So</span> <span m="4556470">we</span> <span m="4556550">can</span> <span m="4556690">think</span> <span m="4556840">of</span> <span m="4556920">general</span> <span m="4557320">is</span> <span m="4557470">M</span> <span m="4557670">and</span> <span m="4557750">N</span> <span m="4557900">as</span> <span m="4558030">being</span> <span m="4558320">arbitrary</span> <span m="4558800">variables</span> <span m="4559730">and</span> <span m="4559940">potentially</span> <span m="4560370">growing.</span> <span m="4560730">M</span> <span m="4560920">can</span> <span m="4561020">be</span> <span m="4561120">a</span> <span m="4561170">function</span> <span m="4561560">of</span> <span m="4561650">N.</span> <span m="4562680">And</span> <span m="4562850">in</span> <span m="4562930">fact,</span> <span m="4563710">when</span> <span m="4563940">M</span> <span m="4564230">is</span> <span m="4564340">the</span> <span m="4564450">square</span> <span m="4564740">root</span> <span m="4564940">function</span> <span m="4565290">of</span> <span m="4565380">N,</span> <span m="4565990">then</span> <span m="4566170">we</span> <span m="4566260">got</span> <span m="4566390">a</span> <span m="4566440">50%</span> <span m="4566960">chance</span> <span m="4567370">of</span> <span m="4567500">a</span> <span m="4567550">match.</span> </p>
<p><span m="4571080">Now,</span> <span m="4573750">the</span> <span m="4573830">birthday</span> <span m="4574150">principle</span> <span m="4574560">comes</span> <span m="4574830">up</span> <span m="4575170">all</span> <span m="4575420">over</span> <span m="4575550">the</span> <span m="4575660">place</span> <span m="4575970">in</span> <span m="4576050">computer</span> <span m="4576370">science</span> <span m="4578700">and</span> <span m="4578980">it's</span> <span m="4579540">worth</span> <span m="4579950">remembering.</span> <span m="4583140">For</span> <span m="4583230">example,</span> <span m="4585090">the</span> <span m="4585190">generic</span> <span m="4585610">form</span> <span m="4585960">for</span> <span m="4586050">this</span> <span m="4586310">is</span> <span m="4586390">when</span> <span m="4586540">you</span> <span m="4586620">have</span> <span m="4586850">a</span> <span m="4586900">hash</span> <span m="4587290">function.</span> <span m="4590510">Let's</span> <span m="4590620">say</span> <span m="4590790">I</span> <span m="4590900">have</span> <span m="4591390">a</span> <span m="4591510">hash</span> <span m="4591940">function,</span> <span m="4593190">h,</span> <span m="4594040">from</span> <span m="4594190">a</span> <span m="4594260">large</span> <span m="4594690">set</span> <span m="4594880">of</span> <span m="4595000">items</span> <span m="4595490">into</span> <span m="4595750">a</span> <span m="4595800">small</span> <span m="4596280">set</span> <span m="4596510">of</span> <span m="4596630">items.</span> <span m="4599280">For</span> <span m="4599420">example,</span> <span m="4599890">say</span> <span m="4600180">I'm</span> <span m="4600280">computing</span> <span m="4600870">digital</span> <span m="4601230">signatures.</span> <span m="4603200">This</span> <span m="4603380">is</span> <span m="4603480">the</span> <span m="4603570">space</span> <span m="4603940">of</span> <span m="4604040">all</span> <span m="4604240">messages,</span> <span m="4605640">this</span> <span m="4605840">is</span> <span m="4605940">the</span> <span m="4606030">space</span> <span m="4606390">of</span> <span m="4606480">all</span> <span m="4606770">1,000-bit</span> <span m="4607620">digital</span> <span m="4607940">signatures,</span> <span m="4609030">and</span> <span m="4609190">h</span> <span m="4609440">is</span> <span m="4609560">a</span> <span m="4609640">digital</span> <span m="4609990">signature</span> <span m="4610480">outcome.</span> </p>
<p><span m="4611750">Say</span> <span m="4612010">I'm</span> <span m="4612130">doing</span> <span m="4612360">memory</span> <span m="4612770">allocations.</span> <span m="4613720">So</span> <span m="4613990">all</span> <span m="4614190">the</span> <span m="4614280">things</span> <span m="4614530">I</span> <span m="4614560">might</span> <span m="4614750">be</span> <span m="4614840">sticking</span> <span m="4615220">into</span> <span m="4616110">a</span> <span m="4616210">register,</span> <span m="4617190">here's</span> <span m="4617430">all</span> <span m="4617550">the</span> <span m="4617640">places</span> <span m="4618000">it</span> <span m="4618030">could</span> <span m="4618190">go.</span> <span m="4618640">Here's</span> <span m="4618870">all</span> <span m="4619050">the</span> <span m="4619330">registers.</span> <span m="4620360">Error</span> <span m="4620690">checking.</span> <span m="4621760">This</span> <span m="4621930">is</span> <span m="4622020">all</span> <span m="4622400">the</span> <span m="4622510">garbled</span> <span m="4622890">messages</span> <span m="4623350">in</span> <span m="4623400">the</span> <span m="4623480">world.</span> <span m="4624490">This</span> <span m="4624580">is</span> <span m="4624670">the</span> <span m="4624760">set</span> <span m="4624970">of</span> <span m="4625030">messages</span> <span m="4625720">that</span> <span m="4625820">make</span> <span m="4625860">sense,</span> <span m="4627360">all</span> <span m="4627600">handled</span> <span m="4628000">by</span> <span m="4628830">functions,</span> <span m="4629630">random</span> <span m="4629940">kind</span> <span m="4630110">of</span> <span m="4630190">functions</span> <span m="4630610">often.</span> </p>
<p><span m="4632770">Now,</span> <span m="4633640">what</span> <span m="4633810">you</span> <span m="4633940">worry</span> <span m="4634320">about</span> <span m="4634620">when</span> <span m="4634730">you're</span> <span m="4634840">hashing</span> <span m="4635420">is</span> <span m="4635560">collisions.</span> <span m="4637830">Let</span> <span m="4638330">me</span> <span m="4638520">define</span> <span m="4638970">that.</span> <span m="4640240">We</span> <span m="4640460">say</span> <span m="4640650">that</span> <span m="4640850">x</span> <span m="4641160">collides</span> <span m="4644850">with</span> <span m="4645080">y</span> <span m="4647680">if</span> <span m="4648290">the</span> <span m="4648370">hash</span> <span m="4648790">of</span> <span m="4648900">x</span> <span m="4649840">equals</span> <span m="4650300">the</span> <span m="4650390">hash</span> <span m="4650790">of</span> <span m="4650920">y,</span> <span m="4652030">but</span> <span m="4652640">x</span> <span m="4652880">and</span> <span m="4652950">y</span> <span m="4653130">are</span> <span m="4653230">different.</span> <span m="4656650">For</span> <span m="4656660">example,</span> <span m="4657180">say</span> <span m="4657420">you're</span> <span m="4657550">looking</span> <span m="4657810">at</span> <span m="4657880">digital</span> <span m="4658210">signatures.</span> <span m="4659930">You</span> <span m="4660110">would</span> <span m="4660310">not</span> <span m="4660670">want</span> <span m="4660900">the</span> <span m="4660990">signature</span> <span m="4661620">for</span> <span m="4662780">a</span> <span m="4663000">$100</span> <span m="4663600">check</span> <span m="4663905">to</span> <span m="4664210">your</span> <span m="4664320">mom</span> <span m="4665620">to</span> <span m="4665770">match</span> <span m="4666140">your</span> <span m="4666280">signature</span> <span m="4667220">for</span> <span m="4667420">$100,000</span> <span m="4668360">check</span> <span m="4669692">to</span> <span m="4670130">Boris.</span> <span m="4672960">Because</span> <span m="4673130">that</span> <span m="4673260">would</span> <span m="4673360">be</span> <span m="4673450">bad</span> <span m="4673770">because</span> <span m="4674030">then</span> <span m="4674180">Boris</span> <span m="4674550">could</span> <span m="4674650">come</span> <span m="4674880">in</span> <span m="4675810">and</span> <span m="4675980">take</span> <span m="4676260">that</span> <span m="4676630">check</span> <span m="4676930">to</span> <span m="4677000">your</span> <span m="4677120">mom</span> <span m="4677370">for</span> <span m="4677680">$100,</span> <span m="4678590">converted</span> <span m="4678855">to</span> <span m="4679120">a</span> <span m="4679210">$100,000</span> <span m="4679930">check</span> <span m="4680260">to</span> <span m="4680360">him</span> <span m="4681960">and</span> <span m="4682160">the</span> <span m="4682220">signature</span> <span m="4682485">is</span> <span m="4682750">authentic</span> <span m="4683730">if</span> <span m="4683890">there's</span> <span m="4684370">a</span> <span m="4684420">collision</span> <span m="4684850">in</span> <span m="4684890">the</span> <span m="4684930">signatures.</span> </p>
<p><span m="4686040">So</span> <span m="4686180">very</span> <span m="4686620">important</span> <span m="4687150">when</span> <span m="4687250">you're</span> <span m="4687320">doing</span> <span m="4687520">hash</span> <span m="4687810">functions</span> <span m="4688200">and</span> <span m="4688590">in</span> <span m="4688720">many</span> <span m="4688940">applications,</span> <span m="4689570">you</span> <span m="4689680">don't</span> <span m="4689980">want</span> <span m="4690180">collisions</span> <span m="4691850">because</span> <span m="4692130">all</span> <span m="4692310">the</span> <span m="4692373">whole</span> <span m="4692436">thing</span> <span m="4692500">start</span> <span m="4692930">breaking.</span> <span m="4694160">Memory</span> <span m="4694490">allocation.</span> <span m="4695020">You</span> <span m="4695080">don't</span> <span m="4695140">want</span> <span m="4695270">to</span> <span m="4695310">assign</span> <span m="4695620">two</span> <span m="4695790">things</span> <span m="4696080">in</span> <span m="4696110">the</span> <span m="4696200">same</span> <span m="4696450">place.</span> <span m="4697770">Error</span> <span m="4698080">correction.</span> <span m="4698880">There's</span> <span m="4699030">only</span> <span m="4699190">one</span> <span m="4699460">answer</span> <span m="4699770">you</span> <span m="4699870">want</span> <span m="4700020">to</span> <span m="4700060">get</span> <span m="4700200">out</span> <span m="4700430">at</span> <span m="4700505">the</span> <span m="4700580">end.</span> </p>
<p><span m="4703100">Now,</span> <span m="4703680">from</span> <span m="4703800">the</span> <span m="4703930">pigeon</span> <span m="4704260">hole</span> <span m="4704400">principle,</span> <span m="4704810">you</span> <span m="4704930">know</span> <span m="4705890">if</span> <span m="4706030">this</span> <span m="4706240">set</span> <span m="4706450">is</span> <span m="4706610">bigger</span> <span m="4706690">than</span> <span m="4706770">that</span> <span m="4707020">set,</span> <span m="4707900">there</span> <span m="4707955">is</span> <span m="4708010">going</span> <span m="4708130">to</span> <span m="4708190">be</span> <span m="4708330">a</span> <span m="4708360">collision.</span> <span m="4708900">That's</span> <span m="4709180">what</span> <span m="4709380">the</span> <span m="4709470">pigeon</span> <span m="4709760">hole</span> <span m="4709890">principle</span> <span m="4710280">says.</span> <span m="4710720">Two</span> <span m="4710900">guys</span> <span m="4711180">will</span> <span m="4711260">get</span> <span m="4711450">mapped</span> <span m="4711660">to</span> <span m="4711720">the</span> <span m="4711800">same</span> <span m="4712050">thing.</span> <span m="4713490">However,</span> <span m="4714010">often</span> <span m="4714410">in</span> <span m="4714490">practice</span> <span m="4715010">what</span> <span m="4715100">we</span> <span m="4715240">care</span> <span m="4715590">about</span> <span m="4716430">is</span> <span m="4716620">a</span> <span m="4716690">subset</span> <span m="4718010">L</span> <span m="4718270">prime</span> <span m="4719810">of</span> <span m="4720230">L</span> <span m="4720530">that's</span> <span m="4720710">pretty</span> <span m="4720930">small</span> <span m="4722160">because</span> <span m="4722370">the</span> <span m="4722450">set</span> <span m="4722650">of</span> <span m="4722720">messages</span> <span m="4723190">we</span> <span m="4723330">really</span> <span m="4723660">assign</span> <span m="4724080">is</span> <span m="4724200">pretty</span> <span m="4724420">small</span> <span m="4725200">compared</span> <span m="4725550">to</span> <span m="4725675">all</span> <span m="4725800">1,000-bit</span> <span m="4726370">signatures</span> <span m="4726765">that</span> <span m="4727160">are</span> <span m="4727410">possible.</span> </p>
<p><span m="4728370">And</span> <span m="4728560">what</span> <span m="4728710">you'd</span> <span m="4728890">like</span> <span m="4729670">is</span> <span m="4729890">that</span> <span m="4730020">for</span> <span m="4730110">this</span> <span m="4730330">smaller</span> <span m="4730810">set</span> <span m="4731010">of</span> <span m="4731100">messages,</span> <span m="4731580">you</span> <span m="4731690">might</span> <span m="4731950">want</span> <span m="4732110">to</span> <span m="4732180">assign,</span> <span m="4732820">they</span> <span m="4733220">all</span> <span m="4733430">get</span> <span m="4733610">mapped</span> <span m="4734400">one</span> <span m="4734710">to</span> <span m="4734810">one.</span> <span m="4736210">And</span> <span m="4736350">the</span> <span m="4736440">birthday</span> <span m="4736800">principle</span> <span m="4737910">says</span> <span m="4738620">life</span> <span m="4738880">is</span> <span m="4739000">not</span> <span m="4739210">so</span> <span m="4739350">nice.</span> <span m="4741560">So</span> <span m="4741700">let</span> <span m="4741770">me</span> <span m="4741840">write</span> <span m="4742060">that</span> <span m="4742260">down</span> <span m="4742710">then</span> <span m="4742960">we'll</span> <span m="4743437">be</span> <span m="4743914">done.</span> <span m="4750592">All</span> <span m="4751069">right.</span> <span m="4751560">So</span> <span m="4751680">the</span> <span m="4751770">birthday</span> <span m="4752170">principle</span> <span m="4763140">says</span> <span m="4763590">that</span> <span m="4765200">if</span> <span m="4765650">S</span> <span m="4765960">is</span> <span m="4766080">at</span> <span m="4766170">least</span> <span m="4766440">100,</span> <span m="4769870">L</span> <span m="4770110">prime</span> <span m="4770500">is</span> <span m="4770600">a</span> <span m="4770670">subset</span> <span m="4771100">of</span> <span m="4771230">L</span> <span m="4772451">that</span> <span m="4772860">is</span> <span m="4773310">at</span> <span m="4773420">least</span> <span m="4774270">the</span> <span m="4774370">square</span> <span m="4774650">root</span> <span m="4774840">of</span> <span m="4774960">S.</span> </p>
<p><span m="4776590">So</span> <span m="4776710">the</span> <span m="4776800">cardinality</span> <span m="4777135">of</span> <span m="4777470">the</span> <span m="4777600">things</span> <span m="4777820">you</span> <span m="4777940">want</span> <span m="4778250">to</span> <span m="4778770">hash</span> <span m="4779740">is</span> <span m="4779880">bigger</span> <span m="4780100">than</span> <span m="4780280">1.2</span> <span m="4781180">square</span> <span m="4781570">root</span> <span m="4781920">the</span> <span m="4782020">cardinality</span> <span m="4782770">of</span> <span m="4783100">S.</span> <span m="4785800">And</span> <span m="4787390">if</span> <span m="4787650">the</span> <span m="4787750">values</span> <span m="4793145">of</span> <span m="4793640">the</span> <span m="4793780">function</span> <span m="4794510">h</span> <span m="4795460">on</span> <span m="4795630">L</span> <span m="4795800">prime</span> <span m="4797290">are</span> <span m="4797490">randomly</span> <span m="4798090">chosen,</span> <span m="4801350">uniform,</span> <span m="4807280">and</span> <span m="4807570">mutually</span> <span m="4808010">independent,</span> <span m="4814260">then</span> <span m="4814500">there's</span> <span m="4814670">at</span> <span m="4814750">least</span> <span m="4815050">a</span> <span m="4815100">50%</span> <span m="4815820">chance,</span> <span m="4816500">so</span> <span m="4816630">with</span> <span m="4816790">probability</span> <span m="4819000">at</span> <span m="4819090">least</span> <span m="4819360">1/2,</span> <span m="4822720">there's</span> <span m="4822920">a</span> <span m="4822970">collision.</span> <span m="4824310">There</span> <span m="4824450">exists</span> <span m="4824850">an</span> <span m="4824980">x</span> <span m="4825250">and</span> <span m="4825335">a</span> <span m="4825420">y</span> <span m="4826350">such</span> <span m="4826710">that</span> <span m="4827590">x</span> <span m="4827880">does</span> <span m="4828070">not</span> <span m="4828310">equal</span> <span m="4828610">y--</span> <span m="4829340">and</span> <span m="4829430">these</span> <span m="4829590">are</span> <span m="4829660">in</span> <span m="4829695">L</span> <span m="4829730">prime--</span> <span m="4831650">but</span> <span m="4832870">h</span> <span m="4833160">of</span> <span m="4833280">x</span> <span m="4834480">equals</span> <span m="4834960">h</span> <span m="4835190">of</span> <span m="4835590">y.</span> </p>
<p><span m="4837840">All</span> <span m="4837970">right.</span> <span m="4838120">The</span> <span m="4838200">proof</span> <span m="4838480">is</span> <span m="4838860">not</span> <span m="4839140">hard,</span> <span m="4839440">it's</span> <span m="4839560">just</span> <span m="4839800">we</span> <span m="4839940">more</span> <span m="4840150">or</span> <span m="4840170">less</span> <span m="4840450">did</span> <span m="4840640">it.</span> <span m="4841910">You</span> <span m="4842010">just</span> <span m="4842210">plug</span> <span m="4842500">in</span> <span m="4843100">the</span> <span m="4843200">cardinality</span> <span m="4843730">of</span> <span m="4843765">L</span> <span m="4843800">prime</span> <span m="4844340">for</span> <span m="4844480">M</span> <span m="4844880">and</span> <span m="4845120">the</span> <span m="4845260">cardinality</span> <span m="4845830">of</span> <span m="4845900">S</span> <span m="4846300">for</span> <span m="4846530">N.</span> <span m="4847620">And</span> <span m="4847810">it's</span> <span m="4847930">bad</span> <span m="4848230">news</span> <span m="4848680">because</span> <span m="4849010">it</span> <span m="4849120">means</span> <span m="4849880">it</span> <span m="4850000">doesn't</span> <span m="4850320">take</span> <span m="4850600">very</span> <span m="4850890">many</span> <span m="4851130">messages,</span> <span m="4852450">just</span> <span m="4852670">square</span> <span m="4853230">root</span> <span m="4853940">the</span> <span m="4854030">number</span> <span m="4854290">of</span> <span m="4854370">signatures</span> <span m="4855560">to</span> <span m="4855670">get</span> <span m="4855840">a</span> <span m="4855900">collision.</span> <span m="4857940">You'd</span> <span m="4858310">hope</span> <span m="4858700">you</span> <span m="4858800">could</span> <span m="4858980">get</span> <span m="4859980">that</span> <span m="4860350">you</span> <span m="4860490">could</span> <span m="4860640">have</span> <span m="4860840">L</span> <span m="4861030">prime</span> <span m="4861310">be</span> <span m="4861450">as</span> <span m="4861590">big</span> <span m="4861810">as</span> <span m="4861930">S</span> <span m="4862330">and</span> <span m="4862730">that</span> <span m="4862780">somehow</span> <span m="4862830">they'd</span> <span m="4862880">all</span> <span m="4863200">go</span> <span m="4863350">one</span> <span m="4863590">to</span> <span m="4863700">one,</span> <span m="4864710">that</span> <span m="4864850">everybody</span> <span m="4865210">in</span> <span m="4865270">this</span> <span m="4865440">room</span> <span m="4865660">would</span> <span m="4865760">have</span> <span m="4865930">a</span> <span m="4865980">different</span> <span m="4866260">birthday.</span> <span m="4866970">That</span> <span m="4867120">is</span> <span m="4867260">not</span> <span m="4867520">how</span> <span m="4867730">it</span> <span m="4867800">works</span> <span m="4868570">if</span> <span m="4868720">things</span> <span m="4868960">are</span> <span m="4869040">random,</span> <span m="4869380">which</span> <span m="4869560">is</span> <span m="4869660">the</span> <span m="4869740">case</span> <span m="4870010">you</span> <span m="4870100">usually</span> <span m="4870480">like</span> <span m="4870680">to</span> <span m="4870770">have.</span> </p>
<p><span m="4871750">Now,</span> <span m="4871860">this</span> <span m="4872050">technique</span> <span m="4872650">is</span> <span m="4872760">used</span> <span m="4873170">to</span> <span m="4873240">crack</span> <span m="4873950">cryptographic</span> <span m="4874600">protocols</span> <span m="4876000">and</span> <span m="4876130">it's</span> <span m="4876260">called</span> <span m="4876530">the</span> <span m="4876590">birthday</span> <span m="4877050">attack</span> <span m="4877990">based</span> <span m="4878290">on</span> <span m="4878330">the</span> <span m="4878410">birthday</span> <span m="4878720">principle.</span> <span m="4879740">So</span> <span m="4879850">what</span> <span m="4879960">you</span> <span m="4880040">do</span> <span m="4880320">is,</span> <span m="4880400">you</span> <span m="4880480">get</span> <span m="4880610">a</span> <span m="4880650">bunch</span> <span m="4880890">of</span> <span m="4880960">messages</span> <span m="4881970">that</span> <span m="4882090">are</span> <span m="4882170">encrypted</span> <span m="4883480">and</span> <span m="4883940">pretty</span> <span m="4884190">soon</span> <span m="4885070">you</span> <span m="4885190">find</span> <span m="4885640">two</span> <span m="4885950">that</span> <span m="4886260">get</span> <span m="4887000">maybe</span> <span m="4887240">encrypted</span> <span m="4887600">the</span> <span m="4887670">same</span> <span m="4887950">way.</span> <span m="4888350">And</span> <span m="4888460">once</span> <span m="4888660">you</span> <span m="4888760">have</span> <span m="4889080">that,</span> <span m="4889790">now</span> <span m="4889930">you</span> <span m="4889965">can</span> <span m="4890000">go</span> <span m="4890260">back</span> <span m="4890700">and</span> <span m="4890950">crack</span> <span m="4891540">the</span> <span m="4891610">crypto</span> <span m="4891950">system.</span> <span m="4893640">For</span> <span m="4893720">example,</span> <span m="4894110">you</span> <span m="4894340">break</span> <span m="4894580">schemes</span> <span m="4894870">like</span> <span m="4895060">RSA</span> <span m="4896210">with</span> <span m="4896350">a</span> <span m="4896390">birthday</span> <span m="4896790">attack</span> <span m="4897680">if</span> <span m="4898610">this</span> <span m="4898780">space</span> <span m="4899250">is</span> <span m="4899370">not</span> <span m="4899600">big</span> <span m="4899770">enough</span> <span m="4900390">and</span> <span m="4900470">that's</span> <span m="4900630">one</span> <span m="4900780">reason</span> <span m="4901070">why</span> <span m="4901570">now</span> <span m="4901770">RSA,</span> <span m="4903040">the</span> <span m="4903150">keys</span> <span m="4904110">have</span> <span m="4904310">thousands</span> <span m="4904760">of</span> <span m="4904850">digits</span> <span m="4906310">because</span> <span m="4906500">otherwise</span> <span m="4906850">you</span> <span m="4906940">can</span> <span m="4907040">use</span> <span m="4907190">attacks</span> <span m="4907580">like</span> <span m="4907740">this</span> <span m="4908720">and</span> <span m="4910020">crack</span> <span m="4910330">them</span> <span m="4910630">more</span> <span m="4910850">easily.</span> </p>
<p><span m="4912450">Any</span> <span m="4912600">questions</span> <span m="4913040">about</span> <span m="4913280">that?</span> <span m="4915760">OK.</span> <span m="4916110">Very</span> <span m="4916380">good.</span> <span m="4916580">We're</span> <span m="4916750">done</span> <span m="4917030">for</span> <span m="4917160">today.</span> </p>
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